I think both terms are appropriate. Actually many mental illnesses do define what the person is like in an overall sense while others don't & are only caused by situations the person it temporararaly dealing with (which may not be so temporary at the time or they wouldn't get the dx.)

I think we need to accept whech ever a person themselves identifies with....

Really, people say "I am gay", not "I have gay" & many people are born with the mental illnesses they have too.