Context-free and context-sensitive grammars in FSM with counters
These counter automata have power incomparable to pushdown automata. In particular they are strictly weaker than counter machines as they have no proper zero test (if zero then .. else ..).
Technically they are known as partially blind multicounter. Blind as they cannot act on zero, partially as they store only nonnegative numbers, and block when a counter is less than zero.
As a language accepting device they are equivalent to Petri nets.
They have been studied for instance by Greibach.