Can you use an Apple Gift Card to purchase Apple Care?

When you read from the first link you provide, it says that terms are subject to change and some items may not be available for purchase. That really leaves it open and you would have to ask Apple if it is permitted.

The second link refers to Apple ID balance, in other words, what your balance is in your iTunes account, which is not with Gift Cards.

Your best source for advice on this is to speak to Apple directly. Use the Contact Support link at the top of the page here to contact them.

Hello,

I understand your frustration. But I want to make clear that I'm in no way an Apple rep so I cannot speak for them. I'd suggest that you contact them to figure the best sulution.

Hai CED86,

You can not use an AppleStore Gift Card to purchase any software, including OS software, or AppleCare.

I hope that this answers your question :)