Capstan and windlass are different devices for getting the same sorts of jobs done. For the biggest job to which they are put, weighing anchor, a vessel would have one or the other and it would be located on the foredeck. Having a windlass for that, however, does not, as far as I know, rule out having lighter capstans located elsewhere on the vessel.

Despite it being possible that a vessel might carry either device, this wasn't a random thing. If one confines oneself to merchant ships, one can see that some or other device was more typical for a certain time period. Windlasses were generally preferred in merchant ships. Once the "handspike windlass" was replaced by the newly invented (1830s) "lever windlass" (pumped), things really took off. The (lever) windlass is what I believe we should expect to be carried by most merchant ships from the 1840s until at least the 1870s -- arguably the prime years of chanty development. It was not as practical to have a traditional capstan for such ships/crews, which were big affairs needing lots of hands.

However -- the exact date is not in my head, let's say "as early as the 1870s / by the 1880s" -- the lever windlass was superseded in many cases by a new device that combined the working principle of a capstan, above deck, with a connected windlass mechanism below deck. This device might be referred to as a "capstan-windlass" or, confusingly, just as a "windlass." The capstan-windlass of later years and the lever windlass of the mid-century thus have the same mechanics in terms of where their power comes from, but on the "user end" -- what crew are doing -- they are totally different. The lever windlass of the exact type, a big one, of the mid 19th century would at some point fall out of use entirely (presumably in favor of the capstan-windlass) except in the case of some small schooners and such with little space, which used a smaller variation of the lever windlass -- the action of which is a bit different.

An issue of course, therefore, is knowing which thing an author might be referring to. Sometimes we get enough description to know. When Clark in 1867 (first published use of "chanty" in a sailor account) describes chanties it is clear he is talking about the lever windlass. Other times we can make an educated guess based on what was the more common in the time period. And just as knowledge of the devices' operation can provide clues about the music, knowing what songs were song could provide clues to historians about the devices in use.

Knowledge of working at the classic lever windlass is effectively "lost." I mean, only people alive in the early twentieth century are likely to have seen it, and all others have no reason to think about it. Currently, these windlasses are like unicorns, so the question is rare to come up. People familiar with the newer (smaller) lever windlass assume that all windlasses worked the same. Clues only come in a few chanty discussions; it's the discussion of music that occasions bringing this up at all. Yet since one of our earliest observed "modern" chanties, "Sally Brown," was observed simultaneously with the newly invented lever windlass (1830s), it behooves us to know how the thing worked. I guess that the vast majority of people giving expositions on chanties, saying were "for heaving and hauling" and impressing their audiences with "starboard and larboard" lingo, have glossed over "windlass" whenever they saw it written in a chanty collection. "Collectors" often lumped "capstan and windlass" in a way that makes sense only if that means "not a hauling song". That's why "everyone" talks about the capstan, the capstan, the capstan, and that's why, I think, some people's sense of repertoire is skewed: They imagine an item as central to the repertoire because they can imagine it sung at a capstan—the problem being that practically anything can be sung at a capstan. (If we want to deduce what *I* at least feel to be the most core, characteristic chanty repertoire, then we have to be able to envision it sung either at halyards or windlass.)

Disclaimer: Some of what I've wrote above is stuff I know intimately and some I merely believe I know "well enough" to advance a discussion towards a point. If my lack of intimate knowledge of the latter has led me to misstate/overstate something, I hope someone will come along to correct me!