Reductions where the number of certificates from one problem can be computed for another to varying degrees
Parsimonious reduction.
Polynomial-time Turing reduction (a.k.a. Cook reduction).
Polynomial-time many-one reduction (a.k.a. Karp reduction).


I've not come across cardinality restrictions of the form you mention in (2) so I don't know any name for them. If you find there isn't a name already, "monotone" would be a good one to choose.  In the case where the number of solutions to&nbsp;$B$ is less than&nbsp;$A$, you need "less than or equal to" since, if $A$ has exactly one solution, you still need to reduct it to a "yes" instance of&nbsp;$B$.