I will just sum up the situation as I see it (too big for the comment box).

One important goal is to set up a good intersection theory for cycles without quotienting by rational equivalence, and using it to get a composition product for finite correspondences, which are by definition elements of groups of the form $c_{equi}(X\times_S Y/X,0)$

It is true that the variety of definitions of cycle groups in the paper is somewhat confusing. There are 16 possible groups because starting from the "bare" notion of relative cycles (def. 3.1.3) there are 4 binary conditions : being effective, being equidimensional, having compact support (c, PropCycl), and being "special", i.e satisfying the equivalent conditions of lemma 3.3.9 (everything except Cycl and PropCycl). So you have

1)$z_{equi}(X/S,r)\subset z(X/S,r)\subset Cycl(X/S,r) \supset Cycl_{equi}(X/S,r)$

and their effective counter-parts.

2)$c_{equi}(X/S,r)\subset c(X/S,r)\subset PropCycl(X/S,r) \supset PropCycl_{equi}(X/S,r)$

and their effective counter-parts.

(1) is then a "subline" of 2))

In a sense, the most satisfying definition would be to use only cycles which are flat over $S$ (the $\mathbb{Z}Hilb$-groups, or the closely related $z_{equi}$) but pullbacks along arbitrary morphisms are not defined there in general.

With the groups Cycl, thanks to the relative cycle condition built in Cycl, you have pullbacks along arbitrary morphism, but only with rational coefficients (thm 3.3.1, the denominators of the multiplicities are divisible by residue characteristics)

The main interest of the "special" relative cycles $z(-,-)$ is in their definition : they admit **integral** pullbacks ! Then you have the small miracle that this condition is stable by those pullbacks and you get a subpresheaf. This means that using them you can set up intersection theory with integral coefficients even on singular car p schemes.

All this zoology simplifies when $S$ is nice : there are some results when $S$ is geometrically unibranch, but the nicest case is $S$ regular, in which the chains of inclusions I wrote down collapse, you are left with two distinctions which are reasonable from the point of view of classical intersection theory : effective/non-effective, general/with compact support. Furthermore, the intersection multiplicities are computed by the Tor multiplicity formula, so the Suslin-Voevodsky theory is really an extension of local intersection theory of regular rings as in Serre's book.