Yes, that's the field. It's technically an assembly of the two field coils (painted red on yours) and the iron/steel core or whatever it's called.
If you say the motor heats up fast, and there were those spots on the field coil that we were skeptical about, it's possible the field coils have an internal short. Get a wattmeter, or ammeter, actually most multimeters have an amperage setting, and run the vac through it to measure the amps/watts, and compare against the vac's rating. If it's way off, you have a problem.
If it's not, well, some motors just behave differently. Likely the smell you have is a combo of ozone from brush arcing, oil and bearing grease, and some remaining dust on the thing. And the spin-down time could simply be the new ball bearing, it does take a bit of break-in.
You sure the motor's not binding? Tried spinning it by hand, and listen for binding?