Michigan State Law says they do not require a fence permit, but my city says I have to have one?
Answers:
In this case, no. Cities and other municipalities (counties, townships, villages) are allowed to pass laws that are more stringent than those of the state, as long as those laws do not violate the State's constitution.
In this case, while there may be no Michigan state law that requires a fence permit, there is a city ordinance that requires you to have a fence permit.
Cities and other municipalities often pass ordinances that are stricter or much more defined than existing state laws to deal with these kinds of situations. A fence permit is something that's much better dealt with at a local level than at a state level.
Other answers:
No, get a permit from the city.
No, get a permit from the city.