Ok, so it's obvious that every organization, dojo, or instructor does it differently. This gives me the perfect opening to vent on one of my small pet peeves. If you are in a system that uses colored belts and go to another dojo and they ask what rank you've achieved, PLEASE do not say something like "I'm a blue belt". Tell them you're a rokkyu or 6th kyu (or whatever it happens to be).
I hear this a lot. People come in and ask about the dojo and say "I studied aikido in college and I made it to yellow belt..." Ok, so what kyu rank is that? "Huh?" Well, what style of aikido did you do, maybe we can figure it out from that? "Huh? I did aikido."
As for the white belts at other schools, I always take mine and try to ask the instructor what they prefer I wear. In one instance I didn't get a chance to see the instructor (whom I knew) before class so I put on the white belt with no hakama. He politetly told me I was being silly and should have worn my normal kit