You are a Michigan resident if Michigan is your permanent home. Your permanent home is the place you intend to return to whenever you go away. A temporary absence from Michigan, such as spending the winter in a southern state, does not make you a part-year resident.
Income earned by a Michigan resident in a nonreciprocal state (see “Reciprocal States”) or Canadian province is taxed by Michigan, and may also be taxed by the other jurisdiction. If you pay tax to both, you can claim a credit on your Michigan return. See instructions for MI-1040, line 18 and the example on page 9.