Am I a NY State resident or part-year resident if I lived in NY from Jan to Aug and moved to NJ in Sep? What does maintain a permanent place of abode in New York mean?

If you lived in NY from Jan to Aug, you are considered a part-year NY resident. Since you did not live in NJ for the full year in 2021, you are also considered as a part-year resident in NJ. 

You need to file at least two part-year resident state returns. 

Your resident state is NJ since it is the state you resided in on 12/31/2021.

If you have a place in NY that you call your home, then you are maintaining a permanent place of abode in NY. If you have no place to call home in NY after you moved, you are a nonresident in NY.