Well that is a bit different, if someone is claiming to be better than others that is a separate argument to someone expressing pride in themselves when they're looked at as less than others in an attempt to be seen as equal.
No, it isn't. Being proud of being in what they see as being the right biological group is the same as being proud of what they see as being the right biological group. A person can only be proud of something they see as being better, either inherently better by birth or as an achievement. It doesn't matter what excuse a person might use to rationalise it to themself or if they're really deluded enough to believe that biological group identity and pride is equality. I've no doubt there are white pride advocates who really believe they're fighting for equality too, for example. There are certainly some who claim to be.
It also requires the belief that a person is their sexuality. You say it yourself right here, when you refer to people expressing pride in being not heterosexual as them expressing pride in themselves - you're saying that not being heterosexual is themselves.
that doesn't seem to be the case, it seems to be mostly motivated by wanting to take a stance against discrimination etc..
There may be some people silly enough to believe that, but it can't be true no matter how fashionable an excuse it is. You don't oppose discrimination on the basis of a biological characteristic by vigorously promoting the idea that that characteristic wholly defines a person and having the "right" version of it should be a source of pride. That's obviously nonsense.