Okay, this is a lot. In order to actually be able to answer it without writing an essay, I won't make complete comparisons for each subrace. As I mentioned in a comment, if you want a detailed feedback, I would create a separate question for each subrace - they are too different from each other.
Your Base Race has almost no features, which is fine considering the main thing is on the subrace. It's still worth to note that consistently the base race is the one giving a +2 AS bonus.
Earth
You get a +3 AS, which is the usual for playable races. +2 Str +1 Cha is an awkward distribution though (same as dragonborn), probably most useful for Paladins (usually the class Dragonborns are played). For most of other classes, one of the two stats will be kinda wasted. By itself, this doesn't make the race weaker. Tiefling gets an even worse bonus distribution.
Barkskin is a 2nd level spell. Not only you are giving a free 2nd level spell to the character, you are giving a free 2nd level spell slot as well. We could enter the discussion if barkskin is a strong spell or not (my opinion: it's not, actually it's weak), but the fact is: 2nd level spells to 1st level players is alot.
I would change it to a feature that might scale with PC's level, similar to Breath Weapon from Dragonborn.
How strong will your resistance to piercing be depends on the campaign. While Dragonborn and Tieflings have stronger resistances (elemental), they are more useful in later stages of the game, while piercing might be too strong in early stages, where physical damage is most common.
I would say the Earth subrace is (maybe too) strong in Tier 1 adventures, but becomes balanced from Tier 2 onwards. I'm not sure about Tier 3, but it becomes kinda weak in Tier 4, compared to the playable races.. One easy way to fix it, as I mentioned, is making the Barkskin feature scale with the tiers. I will make a note here that is valid to the following feedbacks, though: The bonus from races usually are less important in the endgame anyway, since your class features and magic items will compensate for it.
Air
Similar to Aarakocra. I am not sure the +10 move speed in land and Hover compensate both the -1 Dex and -20 Move speed flying, though. Also, I'm not sure how mechanically useful the weather prediction thing will be, that depends on you. Also, while Aarakocra +2 Dex +1 Wis helps alot with Rangers and Druids, the +1 Cha instead makes it a little more awkward, probably being more useful for Bards. Sorcerers might be interested in the flying feature, though, and get more from +1 Cha than +1 Wis.
As it is, it's clearly weaker than Aarakocra. (Yes, I know you asked "compared to PHB", but since PHB doesn't have flying races, I'm forced to compare it to EE, sorry). That said, it isn't completely underpowered since the AS bonus is more attractive for bards and sorcerers.
Fire
My first impression was god it is too strong. Then I noticed that, while you added a lot of powerful features together, they don't have too much synergy.
The main problem is that, from internal comparison, unless Flying can compensate it very well, it's way stronger than the Air version.
Other than that, it's very similar to a Red, Brass or Gold Dragonborn, changing the Str for Dex.
In my opinion, having the Fire Bolt is weaker than the Breath Weapon, even if it can be cast as many times as you want, because you probably will have other ways to deal damage anyway. Thinking in Dex based characters, they will probably be shooting arrows instead of fire bolts. The Breath Weapon is AoE, though, and usually cast by a class that doesn't have that much AoE early (Paladin), so it's more useful in the specific situation it's actually used.
The Grappling damage would also be more useful if it was a +2 Str bonus instead. Building a grappling character without a Str bonus is too suboptimal. I can't even guess how strong this feature will be as it is right now, sorry but you'll have to playtest.
So, comparing to Dragonborns, it will depend on how much Darkvision is a think in your setting. If darkvision is really important, it's balanced compared to Dragonborn.
If we compare it to Tieflings, it seems to get outscaled. Early on, you have a damage cantrip and a grappling feature, while Tiefling has a flavor cantrip. From level 3 onwards they get spells, though, while the scaling of this subrace is only based on the fire bolt cantrip. Again, even if it gets outscaled, it's not an obvious choice since your AS bonus is more attractive to Dex users.
Still, I would fix the fact that the AS bonus (dex) and the features you are giving don't synergize well.
One possible solution is to change the Fire subrace to +2 Str and the Earth to +2 Con instead (if you don't want both to have the same AS bonus). It would fit the "durable" flavor of the Earth/stone and the "aggressiveness" of the Fire.
Note that, if you do, the Fire subrace becomes a very strong option for grappling buildings. I wouldn't say this makes it overpowered, though, since grappling builds are usually more for fun than optimal.
Water
This one is hard to compare. The Healing feature seems weak (just Your Level is not much HP. Compare with Dwarves getting +1 HP/level, which is arguably stronger). Create/Destroy water is a 1st level spell, but this one doesn't need much discussion about being strong or not - it's not strong mechanically.
So it's about the other 3 features. First, the AS bonus: +1 Con +1 Cha is weak. Variant humans get to choose their two +1s and get a Feat on top of that. But they are arguably the best race in the game, so let's not do this. Still, +2 AS is underwhelming and I would expect the features to compensate it strongly. For the Air case, flying might do it, let's see the Water.
Water Breather again will obviously depend on your setting. It's arguably harder to use than Darkvision or Flying, though, since while your party can easily fight in a dark cave or an open field, fighting under water is way more unlikely and usually not desired by PCs.
That leaves Armor of Agathys to compensate the underwhelming AS bonus. Well, sadly it doesn't. It is a very, very strong feature in the first two levels, where +5 HP and 5-10 damage to the enemy are alot, but it quickly becomes weak as you progress in the game. Again, I feel you are lacking scaling. Increase these numbers as the tiers go up. So, similar to the Earth subrace, I'd say it is (maybe too) strong in Tier 1, but gets weaker (and even underpowered) in Tier 3+.
TL;DR
My main criticism here is that all your subraces lack scaling. While Air is fine without it, since Flying itself has a constant utility through the game, the other subraces might suffer from it. It is fine if the campaign is focused in Tier 2, but they seem too strong for Tier 1 and too weak for Tier 3+. Again, note that being too weak in Tier 3+ is not that important, since you have lots of class features and magic items to compensate for that. Still it's something you might want to take into account.
I suggest scaling as a way to fix it, but introducing mechanics that have utility through the game (such as having advantage on something, proficiency on something, etc) also works.
It's worth to mention, though, if you are playing a long campaign that goes through, for example, levels 1 to 15, your subraces are mostly fine, as they will be strong early on and weak later, and it's a valid trade-off for your players to choose. I have linked a chat discussion about this, if you are interested.
Side note:
If you are wondering why I'm not comparing to Dwarves, Elves, Gnomes or Halflings, which also don't have numbers scaling with the levels, it's because they have proficiencies in skills, weapons or tools, or advantages in saving throws, which are the things scaling. Also, they get lots of other features, such as Trance, Luck, etc. Comparing to Dragonborns, Tieflings and Aarakocras is easier and more fair (imo) because they are closer to what OP has made. It's hard to compare if proficiency in perception, using daggers and sleeping 2 hours less is better than 1d8 damage when grappling.
Best Answer
It depends on the campaign, for the "Is this balanced?" question.
I will focus on that in my last section, comparing your race to similar other races.
It's hard to say if features like Flight will be balanced without playtesting it. But, as can be seen in this question, the major problems dwell in early levels. While essentially granting a slower Fly spell for free (opposite to actually casting it, which requires preparing/knowing it and spending a 3rd level spell slot, besides requiring concentration) might seem too strong, it actually depends on how much opportunity you give for the character to use/abuse it.
Most published adventures for Tier 1 (levels 1-4) were (probably) printed assuming the party won't have any ways of flying, thus the flying races are not allowed in AL. From 5th level onwards, the creatures should have ways to deal with flight, as they could be facing Spellcasters with Fly as well.
From that, I would say yes, giving the permanent flight only at 5th level should solve the problems from original Winged Tiefling. How "balanced" it will be, though, still depends on the campaign. As I mentioned, Fly is limited by concentration and spell slots. A 5th level Wizard can only cast Fly twice a day. If there are 8 encounters in the adventuring day where flight helps a lot (i.e., where usually the Wizard would be casting Fly), the feature is still way more powerful than any other in the game. On the other hand, if your encounters are inside a dungeon with a 10ft high ceiling, the flight is worse than having a free cantrip.
From my experience, adventures will have a decent mix between encounters that easily hit flying creatures and encounters that flying will read "This creature is immune to any damage this encounter". In that case, the feature shouldn't be unbalancing/game-breaking. Note that it will still shine more than most other race features.
Note on ASI
Awkwardly enough, I think the major "problem" with your changes is the change in the ASI. The +1 Int for Tieflings, which appears in both the usual and the feral variant, usually meant that the Tiefling only had the +2 primary stat, as Int is a shadowed stat, only used for Wizards and Artificers. By changing that to Dex, you are giving a free +1 "actual" stat for classes like Sorcerer and Bard, which are probably very happy to get Tiefling now.
Comparing to other races
The comparisons, IMHO, are now different than the ones I would usually make for the usual Tiefling (or even the Feral one), due to the ASI changes.
I will focus my comparison with races that have a similar ASI to yours, i.e., Half-elf (+2 Cha, +1/+1 chosen) and Drow (+2 Dex, +1 Cha), since these are the other common races that a Bard, Spellcaster or Warlock would be getting if they weren't choosing your Tiefling, from my perspective. For me, it doesn't make sense to compare between your race and, for example, Dwarf, as I can't see someone actually having to choose between them from an optimization POV.
Comparing it to Half-elfs, you are now trading 2 skill proficiencies, +1 ASI, Fey Ancestry and an Extra Language for resistance against fire and flight at 3rd/5th level. Clearly, your race has better combat applications, exception being for fights against (control) spellcasters (where being immune to sleep and advantage against being charmed are stronger than flying, being paralyzed by Hold Person and taking fall damage). On the other hand, social adventures will favor the Half-elf.
Comparing it to Drows, you are trading Proficiency in Perception, Fey Ancestry, +60 ft in Darkvision and Drow Magic for Fire Resistance, your Flight and not having Sunlight Sensitivity. Again, I consider Fire Resistance more useful than Fey Ancestry, generally, but that might be biased by my adventures experiences (fire damage being more common than charming effects and/or sleep inducing effects). Other than that, the choice becomes essentially "are we fighting outside or in a dungeon?" - outside your race is clearly superior, inside the Drow is clearly superior.
From these comparisons, I say that your race is balanced, as it is not a clear choice over other similar races in every scenario (which is how I read "overpowered"). It might present a problem with very particular scenarios where flight will make that character shine too much, so, as a DM, I would just avoid creating such scenarios, unless specifically for that player and PC feel good about their choice and get some spotlight.