Using "or" to give definitions

If you want to give a definition, or the impression of a definition (because this is not a precise way of doing so) then you have to omit the second "are".

example:
Some people are obnoxious or extremely unpleasant.

If you put in the "are", you are introducing another possibility, another idea. Especially so with the comma before the or.

example:
Some people are obnoxious or are at least widely misunderstood.

"Or" doesn't introduce a definition. "Or" means "alternatively," so in effect you're following with something other than a definition.

You can write, "that is to say," "i.e.".

"Some people are obnoxious, that is to say, they are extremely unpleasant."

"Some people are obnoxious, i.e., they are extremely unpleasant."

I think there is a subtlety here which may have escaped your attention.
I think it can be used in that way, and so the OP is almost on the right track.

You're right. It does lead to ambiguity, though. Given your revision, it can be read as either introducing a definition or presenting an alternative.

Some people are obnoxious or extremely unpleasant.

What do you guys think about setting it off in hyphens?

Obnoxious - or extremely unpleasant - people do exist.

It's okay, maybe a bit too modern for me! It's not necessary from a grammatical viewpoint I think. It doesn't add anything imho.
It's another shot in the locker of the modern author, another tint on the pallette.

That's what I was thinking of. Another syntactical tool. It doesn't add anything substantial to the content, but it's a pretty concise way to get that content across, I think. I do it occasionally, but I don't wear it out. That would be obnoxious.