Let $A$ and $B$ be two decision problems in $NP$. Consider three cases:
(1) For any instance of problem $A$, one can produce, in polynomial time, an instance of problem $B$ having exactly the same number of solutions as $A$.
(2) For any instance of problem $A$, one can produce, in polynomial time, an instance of problem $B$ having a fixed number of solutions for every solution of problem $A$. We can, in polynomial time, calculate the number of solutions for $A$ provided an oracle that gives the number of solutions to $B$, and vice versa. What if we can guarantee that the number of solutions for an instance of $A$ will be greater than the number of solutions for the corresponding instance of $B$ or vice versa?
(3) For any instance of problem $A$, one can produce, in polynomial time, an instance of problem $B$ having a solution if and only if $A$ has a solution. However, we do not have a polynomial time algorithm to compute the number of solutions for $A$ provided some number of solutions for $B$. We also cannot guarantee that such a polynomial time algorithm does not exist.
What formal names and notation for the above sort of reductions? When is a Levin reduction called parsimonious? Instances one and two only? Is the third example necessarily worth anything?