The races are mostly fine
I would replace Tsunami with Control Water for the Neptunian. An AoE spell that does half damage on a successful save is mediocre for a Druid, but better than great for a Fighter1.
The same is true for Lightning Arrow for Jovian, make it once per long rest.
Olympian
As David Coffron wrote, death saving throws should not matter much between magical healing and good encounter design, and Thaumaturgy is mostly for flavor, does not add much to power.
Base race is fine.
Minervian
While two +2 are very rare, only the Mountan Dwarf has it, this combination is much weaker, basically the second worst after +2 Wis / +2 Cha.
I would much rather pick +2 Wis / +1 Con (or Dex)2, so you are way behind a Half-elf.
This also makes the two skills fine.
Legend Lore has expensive components, and is not combat related. It does not add much to your power, while flavorful.
Hermian
Again, way behind a Half-elf for anything that needs Cha, but I could make a decent Cleric or Wizard with it. Still, a Variant Human is better.
Overall interesting and versatile, but far from strong.
Jovian
Looks really similar to the Fire Genasi, you should tie the spells to an ability.
We have a much earlier fly speed precedent with Aaracokra, and Control Weather is mostly for flawor, so this is fine too.
Neptunian
Very similar to Jovian, while Shape Water is significantly weaker than Shocking Grasp. Tidal Wave is comparable in damage to a 16th level Dragonborn's breath, but they get to do it once per short rest. I think it is ok, considering how mediocre the rest is.
As mentioned at the beginning, replace Tsunami with Control Water.
1) the expected damage of an 8th level area spell is 45,5 according to the DMG, but Tsunami only does 33 in the first round, even worse on subsequent rounds. A Druid can do better damage against mobs of weak enemies, and use his concentraion more effectively, but these weaknesses do not apply to a Fighter
2) You mostly need only 3 abilities: Attack, AC and Con. Some classes can merge the first two, but no-one needs both a high Int and Wis.
The Elementalist Fighter class is more powerful at every level than a standard Battlemaster Fighter.
I'm using the Battlemaster Archetype as a reference point as the OP has stated that it is an influence on the homebrew class design and because the class features closely mirror that of the Battlemaster.
TLDR; Elementalist Fighter blends almost all of the features of a Battlemaster, adds a sprinkle of Barbarian and then adds a tonne of powerful abilities to boot.
- Second Wind comes 2 levels later, decreasing survivability at 1st and 2nd level, before returning to the status quo at 3rd.
- Weapon Bond is almost identical to the Eldritch Knight feature and comes 2 levels early. It somewhat undermines one of the EK's unique features but is not too major.
- Imbue Weapon is a reskinned Barbarian Rage, the quintessential Barbarian feature. It is more offensive focused than defensive. There doesn't appear to be an end condition, but I'll assume it's the same as Rage. Almost everything here feels out of flavour for an elemental focused fighter. Psychic resistance, temporary darkvision, changing the advantage on checks and saves to Charisma, and advantage on saves against being charmed are not, in my opinion, things I would associate with a class called "Elementalist Fighter". Mechanically, choosing the additional damage's type is strong as it somewhat enables low-level melee fighters to circumvent physical damage resistances, one of their greatest banes. It is somewhat comparable to the 3rd level Elemental Weapon spell, minus the to hit bonus and magical weapon consideration. Losing the physical damage resistance of Rage is not a suitable drawback, as Fighters are much more capable of achieving high ACs at low levels than Barbarians, through heavy armour, shields, and fighting styles. So the loss of defensive abilities is negated by the classes intrinsic high AC and the damage capabilities are greater than the standard Rage, with some odd extras thrown in.
- Maneuvers. Having extra restrictions on maneuvers is sure to only cause confusion. The save DC being Charisma based is also quite unexpected and again outside of the classes flavour. Some of the new maneuvers are far above the current power level. Elemental Slam stands out as particularly egregious. Fighters lack AoE as a balance requirement. Giving them access to AoE which does good damage for a relatively low resource cost, is definitely going to cause them to be overpowered relative to another fighter of the same level. In conjunction with the powerful secondary abilities that many of the damage type Slams offer, I don't know why you wouldn't pick this maneuver over any other.
- Ancient Blade doesn't have any real balance concerns. It does, however, cause a great deal more work for your DM, which is something that I would consider poor design. Know your Enemy of the Battlemaster is pretty whatever, but it doesn't cause any additional preparation or forethought that could be used on the game as a whole, instead of one character.
- Elemental Incarnation is absolutely insane. Indomitable is already a strong class feature, and EI blows it out of the water. It evaluates to casting one of the Investitures of X spells, without the need for concentration, once per day, at the cost of one level of exhaustion. And this is in a non-casting class, 2 levels earlier than a full caster could. The sheer damage output and/or utility that this feature could provide in its 10 minutes is incredible. Especially considering most conditions that require a save also give one each turn. Potentially missing a turn or two is well worth this.
- Elemental Explosion is again powerful because it gives Fighters access to powerful AoE at a relatively low cost.
If given the choice to play a Battlemaster or an Elementalist Fighter, the Elementalist is the clear winner. With its access to cheap AoE and additional damage output, it does everything a Battlemaster does and more. It suffers from a lack of identity, borrowing iconic features from other classes, and also having a mix of features that don't resound with the classes core concept.
Best Answer
Is this balanced against original Barbarian paths? No. Not at all.
Let's take this level by level:
Level 3: Building Charge
First off, it's unclear as to how the static points are generated. Is it just if there's an enemy in range? Is it if you strike an enemy in range? Regardless of how the points are gained, at level 3, they can have a maximum of 3 points, meaning they can expend either 1 or 2 points to use an action to shock an enemy. This means they'll likely be taking 1d8 or 2d8 lightning damage, which, at earlier levels, seems on par with default barbarian damage.
However, compared with the level 3 abilities from the Player's Handbook, on pages 49 and 50, we run into an issue that will continue to come up throughout this path: the other abilities tend more towards support abilities than damage. The Berserker gains the ability to go into a frenzy and take an attack on a bonus action, but with the harsh penalty of gaining a point of exhaustion at the end of the rage. The Totem barbarian gains either resistance to more damage, more mobility in combat, or an aura that helps nearby allies.
I think at early levels, this ability might not get used; I'm not great at statistics, so if someone has run the math and can tell me I'm wrong, that's fine, but I believe that for a while, this ability will be worse than the barbarian just attacking normally. Once you start reaching higher levels, this starts to be a lot of d8s of damage that the barbarian can put out. And, notably, this is a magical form of damage, which means that the barbarian will be able to effectively hit monsters that are immune to piercing, slashing, and bludgeoning damage. Whether or not this is unbalanced depends largely on the campaign, but it's worth noting.
Level 6: Fly like the Wind
Meh. Flying is incredibly situational. However, the Totem path doesn't get flying until level 14, so this is still weird. Getting flight at level 6 when most classes can't get it until much later is odd.
Level 10: Lightning Strike
This is similar to the Paladin's "Improved Divine Smite" ability, which is gained at level 11, but also does 1d8 damage as opposed to the 1d6 for Lightning Strike. So, compared to a Paladin, this is okay, but again, it's very different from the PHB options. The Totem path gains the commune with nature spell, and the Berserker gains the ability to inflict the frightened condition, both of which I would classify as "support" abilities, rather than damage dealing ones. This is a decent chunk of extra damage, and at level 15, it'll actually outpace the Paladin's Divine Smite.
Level 14: Electric Discharge
Absolutely not. At level 14, this is 7d8 lightning damage to everyone nearby. At level 20, it's 10d8 lightning damage to everyone nearby, in addition to the fact that at level 20, you have unlimited rages. AOE damage is an enormous gamechanger; there's a reason that AOE damage is pretty rare. Having it be once per long rest helps, but it still doesn't bring this on par with the other Barbarian options at level 14. Again, they don't add damage to the class, but rather tactical and supporting abilities. This is the appropriate level for flight, for example, in the PHB.
Finally, regarding intent:
You say your goal is to have the barbarian be able to do damage at range, but I don't think this will work well for that. The Stormy Aura requires lots of targets within 10 feet in order to charge up points, which I wouldn't consider as being at range. Additionally, the point of the Barbarian is to be in melee. That's why they have such high constitution: they're the front line tanky people who stand there and get hit so that the squishy rogues and wizards can toss daggers and spells around without fear.
I think this is a really cool concept, but I think it's weirdly balanced throughout levels. Compared to the original barbarian paths, the extra damage output of this is just too much, especially when the barbarian paths contain primarily support abilities, and the few abilities (like frenzied rage) that grant extra damage come at a high price.