they were still students though, they were not trainee X-men at that time, it wasn't untill X3 that they were
like there was no sign of kitty in X2 (unless you count as tiny cameo) so they just threw her randomly into the mix in X3 as a trainee
They were X-Men by the end of X2 that was why they were in suits. There's no question they had already joined the team. They wouldn't have gone on a mission to the white house otherwise.