What terms to differentiate between a quasi-synchronous motor (eg an induction motor ) and a completely asynchronous motor (eg a universal motor)?
Please understand, electrically and magnetically


synchronous machine and brushed DC machine are twins
asynchronous machine and transformer are twins


(The repulsion motor is a special sort of asynchronous machine. The brushless DC motor is a special sort of synchronous machine.)

So your categories make very little sense. That's why there isn't a term for it.

In case you wonder, the brushed DC machine is synchronous to its own speed, that's what the commutator is for. Because it's always synchronous, you can feed any frequency to it, given that the field current is in phase with the rotor current. In an universal motor, that's ensured by putting the field and rotor winding in series.