I agree that in most contemporary locales these terms would probably now be referring to the same basic type of devices, that could be used in various fashions to tap branch lines into a main line (of course many types of “tapping sleeves” in plan view regardless of materials look very much like a “tee”, and in profile view look like a tee cut in two halves at the horizontal pattern “parting line”). I feel however this unfortunately might not conclusively nail down exactly what you have in the ground from at least old project(s), as I believe there have been many different devices, with at least slightly different names but that could easily be confused, used over the decades in some areas. There was e.g. a “tapped tee”, which I believe was basically a short bell by bell pipe casting with one or more bosses (maybe on each side?) that could be installed either initially or later cut-in to a mainline later with a sleeve (used e.g. often to connect generally smaller threaded steel pipe to the main line). There at one time were also “cutting-in tees”, that basically looked like a tee, but with one socket end of the run instead oriented at an angle to the axis of the run (that allowed for “buck-in” assembly/without a sleeve of the new branching tee, after a precise length spacer/gap was cut in the existing mainline). Unlike most tapping sleeves (or pre-installed tapped tee bosses?) cutting-in tees could not be used to make wet or live taps (e.g. without shutting down the main line). Finally, there were/are of course also tapping “saddles”, that are used very much like tapping sleeves but I think most folks would identify normally employ some sort of brass/bronze or alloy steel etc. “straps” (instead of enclosing halves) to fasten/seal the new branching outlet to the mainline (after assembly, however, these assemblies of course would also look much like a “tee” sort of intersection).
