RAW there really is only one option to quickly take off armor: have someone help you.
Getting Into and Out of Armor
The time it takes to don or doff armor depends on the armor’s category.
Don. This is the time it takes to put on armor. You benefit from the armor's AC only if you take the full time to don the suit o f armor.
Doff. This is the time it takes to take off armor. If you have help, reduce this time by half.
The fastest a PC could shed heavy armor with help would be in 2 and a half minutes or 25 rounds of combat.
RAW options to avoid the damage
- Run away. As I interpret the spell you would still need to be in its range to use the bonus action attack. Running out of range would be difficult to achieve since the spell attack is a bonus action allowing the caster to use both their actions to follow you. However if allies impede the movement of the caster or your move speed is greater it could be accomplished in one round.
- Deal Damage. Every time the magic user takes damage while concentrating on a spell they are forced to make a CON save to keep that spell up. If everyone hits the wizard odds are pretty high that he will fail one of those CON saves.
Houserule option
Let the PC take it off with a standard action, but doing so damages the armor, preventing them from wearing it again until they have it repaired.
Musings on Heat metal
There is no requirement for a PC or NPC whose gear is affected by heat metal to react taking off their armor. The stipulation of the spell is to throw it if they can, donning and doffing seems to be a bit more than all that and as such I would interpret it to mean weapons should be thrown, but armor can be kept on. I know 2d8 bonus action damage sounds like a lot, but dice only damage has a way of only coming out to be around the average most of the time, while 9 damage a round without an attack roll is pretty nice, it is probably a better idea for the party to focus-fire on the wizard to end it rather than someone in heavy armor taking their armor off and seriously lowering their AC for the rest of the fight.
The Rules Don't Say
As nvoigt pointed out, the PHB describes chain mail and plate mail as complete items. They are either on, or they're off. There are rules for how long it takes to put them on and take them off, but there is no description for your AC while wearing just the padding.
Furthermore, the padding under plate and chainmail isn't padded armor. Padded armor is a different thing. When you buy plate mail you don't get plate mail armor that has padded armor inside, you get plate mail armor that has padding inside.
You Could Try It
Since there are no rules that describe this situation, you are down to the most general rule of all. At the beginning of the Players Handbook, it says:
1.The DM describes the environment. ... 2.The players describe what they want to do. ... 3.The DM narrates the results of the adventurers' actions. (PHB p.6, "How to Play")
So, your character could just try it and see what happens. "Try it" could mean a lot of things. "My character takes off all his armor but the padding. What's his AC?". "I'm looking at the swamp we're about to cross. I'm an experienced fighter, I've been wearing armor for years, I've been trained by experts. I know if I fall in wearing full plate, I'll probably drown before I can get out of it. What does my training tell me about whether I can wear just the padding in order to mitigate the drowning risk but still retain some protection?" Or perhaps your armor wearer might consult armor experts.
You've Solved My Drowning-in-Armor Puzzle!
As a GM, I view the water-armor situation as a challenge or problem for the PCs. Which do you do, take off the armor and carry it thereby lowering your AC, or wear it and risk drowning?
Only it isn't an either-or, which is the beauty of RPGs. There are other solutions to the problem. Perhaps the water is such and the PC's strength is such that the PC can just walk out, perhaps with a strength or constitution check. Perhaps the PCs have access to magic that allows them to breathe water. Maybe they can bypass the swamp. The players might (and hopefully will!) have innovative solutions I haven't thought of.
If the PC said, "I take off most of my armor, and wear only the padding", or "only the helmet, gauntlets, greaves and leg guards" and subsequently fell into the water I think the time to remove the armor and thus the risk of drowning would be less, while the chances of swimming in part of the armor would be more. And if the PC were attacked by Rodents of Unusual Size while crossing the swamp, I'd definitely have to allow that the PC's base AC in padding or other bits and pieces was greater than without armor. Wearing only part of the armor might be a reasonable solution to the but-I-might-drown problem. Although wearing only the padding might be really bad for it, given briars and such.
Given that I don't want to play "Armor & Anoxia" I'd be reluctant to come up with a codified houserule that says you can wear half the armor for half the AC and half the risk of drowning, but perhaps that would be the solution. I'd definitely want to keep things liquid, and not weigh the game down with heavy armor questions, but I'd definitely want to pad it enough to keep things moving forward, while at the same time rewarding innovative role-play and problem-solving.
Best Answer
The rules don't cover this, so the DM must make a ruling.
Doffing heavy armor takes five minutes, and that is the highest resolution ever given to the act of doffing heavy armor. The rules do not cover removing individual pieces of complex armor, so when the DM rules that the Knight can remove his gauntlets and helmet within the action economy of a single turn, that is a ruling the DM has made because the rules don't cover the Knight's chosen course of action. It falls under this clause of the Actions in Combat section of the combat rules:
Since this course of action is not covered anywhere in the rules, just as the DM had to rule on the attendant action economy of the course of action, the DM must make a ruling in the further implications of allowing the Knight to take this course of action.
I know this isn't a satisfying answer, in the sense that I provide no guidance to making the ruling, but that's how it goes sometimes. The three rulings you provided all seem entirely reasonable to me, simply because the rules don't speak to this situation.
And finally, this sort of conundrum is at least anticipated by the authors, as we see in the introduction to Dungeon Master's Guide: