The following option exercise style is somewhere between American and European:
There is a fixed grant date $N_1$ at which you determine at which date $N_2>N_1$ the option will be exercised.
So once the grant date $N_1$ arrives, the option becomes European, i.e., you have to commit yourself to when the option will be exercised. But the choice of $N_2$ may depend on the history of stock prices up to time $N_1$.
Perhaps this is Bermudan or Verde. Is there a standard or succinct way to describe it?