Can an interrupted long rest be treated as a short rest?

[Tanarii]

You absolutely can tell the DM "we take a long rest" or "we're going to take a short rest;" I've seen it in play plenty of times. Doing so would prevent later invoking the other, as again, they're mutually exclusive.

I'm not sure why you think it's not possible for players to declare what type of rest they're planning on taking.

You can certainly use it as short hand it for declaring how much time you want to take. But that doesn't lock you out of anything. The benefits accrue from what your character has actually done, not what they declared they were going to attempt to do.

Again, this isn't a video game where the players push the Long Rest or Short Rest button to gain the benefits. If they do the thing, that's when they get the benefits that apply.

Edit: Relevant to topic:
http://theangrygm.com/hitting-the-rest-button/