In Hom's paper (arXiv link), p.20, Section 3.3 ends with

"There are other algebraic modifications one may consider, such as setting $U^n = 0$ or $UV = 0$",

referring to the knot Floer homology with $\mathbb{F}[U,V]$ coefficients. That is, we consider pseudoholomorphic disks in the Heegaard diagram (or rather, in the symmetric product of the Heegaard surface), including those passing through both basepoints $w$ and $z$.

However, there is no reference to works that do consider these different variations. My first question is, which modifications have already been considered/published, and for each what is a good reference?

The second question is similar to the first: in the specific example that we set say $U^2=0$, is there a good geometric interpretation of what kind of information we're losing?