I spose your both right, maybe they should be called semi synchronous.
Current for the rotor starts to flow when it slips, ie the rotor goes slower than the rotating magnetic field, the more slip the more current and the more torque, you could probably measure the load on the motor by its slip.
An induction motor's speed can be and often is control by a drive, the drive changes the frequency to change the speed, so their is some synchronicity.
I've had issues with 60hz motors running on 50hz, standard efficiency ones get hot, most likely the same principle as a transformer, core saturation.