Drinking/administering a potion takes an action, as you cited from DMG at p.139. That's the general rule in effect, unless specifically contradicted.
You note that the DMG's description of the Potion of Healing doesn't mention that it requires an action to consume, and seem to be wondering if that omission is meant to signal something. But every other potion described in the DMG also omits any description of the mechanics of drinking potions: they just start with phrases like "when you drink..." or "for one hour after drinking...." So we can't read anything into the omission, or the general rule would apply in no cases. (DMG5e pp.187-8)
So why does the PHB call out the consume action in its Potion of Healing description?
The Potion of Healing is the only potion described in the PHB (p. 153); the PHB doesn't have a general "potions" section akin to the DMG's on p.139. So this is the natural place in the PHB to mention that interaction with the action economy1. Would it have been clearer to add to the sidebar on p.190? Maybe, but that's a sidebar full of free interactions, not examples of things that aren't-free-interactions-though-you-might-think-they're-just-the-same.
Sidebar: But I can quaff an entire flagon...
Yeah, this bugged my table too. We made in-world sense of it by saying that a magical potion is an... unusual enough mouth-experience to take a moment2 to consume. A flagon of ale... apparently the game designers assume our characters are more-inured to that experience than to the experience of drinking spells.
1 - it's also mentioned in the Herbalism Kit, but that's the least-natural place to talk about the potion's consumption.
2 - I think about trying to get my kids to take emoxicillin and Wish it only took six seconds =P
Whatever works in your game is what you should do. If you haven't tried any of these yet to determine what works though…
Your first two options I've never seen work. They fail exactly in the ways you predict: the piecemeal benefits are way better than the normal function and it fundamentally changes the role of potions in the game. They don't work, for a value of “work” that means “potions have their expected utility.”
What does work is every option after. Your predictions about how they would fail don't bear out in experience.
Descriptive effects take a little bit of creativity to come up with for each type of potion, but it's less difficult than ad-libbing royal dialogue or coming up with a way to respond to players doing something you don't have a plan for.
Yes, a sip of healing potion could be described as “refreshing”, and that's pretty easy to come up with. There's no reason that you can't come up with subjective effects for any other kind of potion, whether or not it's an all-or-nothing effect. An levitation potion might cause one to feel “bubbly”, an invisibility potion might make the drinker “feel light-headed”, a potion of fire breathing might taste “spicy”.
The point is, no matter what you come up with, the hint will not 100% tell them what the potion does anyway. That means that you can get slightly poetic about the descriptions too, so that maybe a potion of growth makes the tester feel “achy” and a potion of water breathing feels “cold”. If you want it to be obvious, then giving hints is defeating your goal; skip to the bottom and read up on another option you didn't think of…
Recognising its taste is perfectly valid. Your concern that it might get old fast doesn't bear out — in practice what happens is that the players get very interested in keeping good notes on what potions taste like so that they can quickly and accurately identify future potions, as well as sort them into safe “known” and risky “unknown” categories. This kind of behaviour is wonderful and exactly what making players experiment with magic items is supposed to elicit. If you're wanting your players to vigorously engage with potions in your game world, this is a boss method to do it.
Recognising its appearance works just like taste, except it's the kinder, gentler option because they don't have to risk imbibing something that might be harmful even when only sipped. Players will take meticulous notes (or curse themselves for not doing so, the first time they find a potion they recognise but can't remember the effect of), and really get into the process of experimentation.
These three can even be combined for even more in-depth experimentation play experiences: give each potion a descriptive sensation when sipped, an appearance, and a taste. You can make these each unique, or you can make some of them the same; e.g. maybe two potions are both a brown oily swirl in the bottle, but the one that tastes like onions is a healing potion and the one that tastes like acetone is a fire-resistance potion. I improvise these details when a new potion is discovered, and I keep a blank spreadsheet page in my notes to jot down the potions' attributes to keep it consistent. (There's nothing stopping you from determining this stuff ahead of time, but I like to save my prep for things that can't be done improvisationally or with random tables during the game, so the blank spreadsheet is how I roll.)
Some potions will be easier to identify than others by different means, which is fine — such variations are the stuff that make small details like potions interesting to interact with in the exploration mode of play. Descriptions of potions they haven't found yet also become valuable information that NPCs can give the PCs, or which can be found in scribbled notes in the margin of a spellbook. Knowledge becomes treasure that the players really value!
Another valid option: Don't make them guess at all
Another option is to handle it in a way similar to describing NPC's conversation in third person: require the exploration action still (and therefore the associated risks of choosing to sip or not sip strange liquids) but skip describing the details and just tell them what their PC learns.
You take a tiny sip from the bottle of liquid and discover that it is a potion of strength. Moving along now…
If the above options that involve the players having to treat potions as mini-riddles — and maybe getting it wrong sometimes — is something you don't want to ever deal with, then you don't actually want them to be guessing via hints. You just want to tell them. And that's fine!
It doesn't even remove the exploration focus from a game — it just changes what the game focuses on exploring. If a player can have their PC just sip a potion and be informed what the hero deduces it to be, without needing to do the deduction themselves, that just means you can move along quickly to the other awesome exploration stuff: mapping, managing light sources, making semi-informed risk assessments about going further or retreating, etc.
Best Answer
I am a huge dumb nerd so I just went and proved this for myself with a bathtub and a bottle of Red Guitar. By pulling the cork with my teeth and immediately folding my thumb over the lips of the bottle, I managed to leak only the merest iota of wine. If I had the clearance to stand upright instead of kneeling there, like a moron, dunking my head and holding the bottle sideways, then the only leak would have been from the margin of air at the top of the container.
Having replaced the seal with my finger, it was then no effort to simply wrap my lips around that of the bottle's, extract my thumb, and admit air from my lungs to equalize the pressure and allow a couple of sips. You can do this yourself without getting your head wet or finding a beverage with a cork.
So my answers is yes, this is a trivial exercise. Even asking for a DEX check would be too much. Only a small child would have a particularly difficult time with this. You might as well ask for a STR check to extract your foot from your shoe.
You may be wondering, "Eikre. Did you seriously get up and run a bath just to have citational highground over the armchair theorists?" All I have to say for myself is that there is a distinct possibility that the sangria I drank in the course of the experiment was not exactly the first sip of the day.