I believe that theism is a specific case of deism, because theists do believe in a god, like deists do. They just also go further and believe that this god is a personal god and does miracles, answers prayers, etc. So, then, is theism a specific case of deism, or different from it?
As Conifold and Brian have pointed out, historically the order is reversed. However, conceptually, you are actually correct (relatively, at least): as Kant worded it, the deist believes in a God, the theist in a living God, and by fleshing out the phrasing (even if with but one more word!), the reporter of theism does render a more specific term than the reporter of deism simply-put would.
Now, I qualified my desire to confirm your suspicions by that word "relatively," so I would add that the two descriptions, "A God who is not occurrently active in the world," and, "A God who is so active," are perhaps not necessarily more or less general than each other. There's some naive Occam's-Razorism to the inactivity case, granted, so one might compare the deist to someone who believes in an empty set but no full sets (so to speak), but then is an empty set more general than any full one?
I would reverse that, and say deism is a subset of theism. Theism is closer to compulsory law, whereas deism is more advisory. The founders of the USA were self-proclaimed deists who believed in a creator whose law was above any statutory law. That is, rights do not originate from government. Government exists primarily to protect the rights of the weak from violation by the mighty.
John Locke, "Two Treatises of Government"