Caveat: never personally used ms, I might be missing a subtlety of how this model is intended to be used.
I think it's useful to see how Hudson himself writes about the utility of the infinite-sites model in the context of finite recombination. Recall, infinite sites is a simplifying assumption to make math easier. Tajima (1996) boils this down as follows:
...the infinite-site model proposed by Kimura (1969) assumes that the total number of sites in each gene is so large and the mutation rate per site is so small that whenever a mutant appears it occurs at a previously homoallelic site.
Thus, infinite sites just says (among other things) that any given mutation is at a new site where no new mutation has occurred previously. This is an assumption that is handy for doing population genetics, which Hudson employs. In one paper he (and a colleague) write:
HUDSON (1983b) has also developed an efficient method for simulating samples from the neutral infinite-site model with finite recombination. His method generates the "history" of a sample. The history of a sample is a collection of correlated family trees, one for each site (for DNA sequence data, each nucleotide is considered a site). The family tree for a site traces the genealogy of a site back to its most recent common ancestor indicating which sampled gametes are most closely related and when the most recent common ancestors occurred.
If the rate of recombination is zero, then each site has the same family tree and therefore the history of the sample consists of just one tree. The method for generating this tree depends on results of WATTERSON (1975) (for details see HUDSON 1983; TAJIMA 1983). On the other hand, if the recombination rate is infinite, then all of the family trees are independent of each other, and each family tree is generated in the same way as when the recombination rate is zero. If the recombination rate is finite, then the topologies and lengths of the branches of the Family trees are correlated because of linkage, and generating them is more complex but still possible (HUDSON 1983b).
In other words, infinite sites is an assumption that helps us study the behavior of recombination (which is, of course, finite). We could alternately consider models in which we make zero or infinite assumptions about recombination, to study the behavior of finite mutating sites.
Elsewhere in the documentation for ms, Hudson writes:
In this model, the number of sites between which recombination can
occur is finite and specified by the user (with the parameter nsites). Despite the finite number of sites between which recombination can occur, the mutation process is still assumed to occur according to the ”infinite-sites” model, in that no recurrent mutation occurs, and the positions of the mutations are specified on a continuous scale from zero to one, as in the no-recombination case.
So it seems that he is using the property of no recurrent mutations laid out by Tajima, but also the feature of assuming that coordinates of the region of interest are continuous rather than discrete. One can imagine that this is convenient to make computation simple.
I think that the mental model that you've laid out is not bad in terms of an intuition about behavior, but to emphasize the main properties that I think are useful, one might say that 1) every mutation/allele is in a different (continuous) coordinate (no recurrent mutation), and 2) the number of possible recombination sites is discrete (nsites). So there are nsites recombining regions, each without recurrent mutation (and possibly infinite continuous sites, though I don't see how this is actually implemented so it may simply be that each of a finite set of sites doesn't recurrently mutate).
