Are coral cays and coral atolls just a different type of volcanic island? When I hear: "volcanic island" I think at places like Moorea or Hawaii, but ultimately, coral cays and coral atolls lay their foundation on the residuals of a volcano, right?
No, they are not necessarily volcanic.
Both of these terms, along with many others like fringing reef and barrier reef, are just morphologic — to do with shape. They tend not to imply anything genetic — to do with origin. (The separation of description and interpretation is an important principle in observational sciences like geoscience.)
Cay A low island, often of sand or coral. See Wiktionary.
Atoll A ring-shaped coral reef. See Oxford English Dictionary.
The name Cay is sometimes applied more generally to an island or group of islands. In some cases, as in Cay Sal Bank, Bahamas, the feature is actually an atoll: a ring-shaped reef. Cay Sal Bank is one of the largest atolls in the world, and did not form on a volcano.
Charles Darwin hypothesized in his monograph The structure and distribution of coral reefs that atolls form when a fringing reef around a volcanic islands grows as the volcano subsides:
Indeed, Darwin turned out to be correct that atolls often have a volcanic origin, but Cay Sal Bank is a counter-example. I think it's fair to say that most people would assume that origin though, and lots of Internet definitions , so it's notable if an atoll formed some other way, e.g. on a non-volcanic seamount (see Keating et al., 1987, in Seamounts, Islands, and Atolls, AGU), or by island subsidence due to dissolution.