But to me, "move" and "migrate" mean different things.
If Tim moves to the United States, his citizenship may not change but "migrate" does.
Am I right?
I see your point. Someone who moves to a new country with the intention of staying is an
immigrant who is
immigrating. Such people have
emigrated from their previous home country and can be called
emigrants from it.
It would be tidy and logical to call them
migrants, but we usually don't.
It usually takes a long time to establish citizenship in a new country. A person legally living in a new country with the intention of staying is called a
permanent resident in the United States and a
landed immigrant in Canada (and, I suspect, other Commonwealth countries).