"The "Funky Memphis" drum sound was probably several things. As far as the drums actually recorded in Stax, of course the biggest part of the sound would have been Al Jackson, Jr., or in some cases, Willie Hall ("second string drummer from The Bar-Kays, later with The Blues Brothers). Secondly, the drums of course were important. Al used a Rogers kit mostly, and a Ludwig 400 snare. Those snares are amazing. The Stax drums were recorded in a pretty small drum booth. Al didn't have a lot of extra room in there. Also, that would mean very little "room" sound as we know it today. Mic'ing was pretty minimal. Usually a KM-84 (NEVER a 184, please, they are not the same!) on snare, but not as close mic'd as normal today. Perhaps about 5-8" and pointing at the snare. The hat was almost never mic'd. One or two 87's or 67's (usually one) would have captured the "rest of the kit" most of the time. The bass drum might have an RE-20 on it, or on occasion, either the snare or bass drum might have an RE-15 or a Shure 545. That's about it. Usually, bass guitar (mostly passive transformer direct), bass drum and snare were tracked together on track one, the "rest of the drums" on track two, and then the other music on any other tracks. The console (never any outboard pre's) was either a Fluckinger or a Spectrasonics, and recording would have been to Scully machines.
At Hi Studio, where they did Al Green and some other great ones, the drummer was....Al Jackson again! The setup was somewhat similar, although the console distorted quite a bit, and there was always a conga player playing along (Howard Grimes), "doubling" the bass drum and snare, and playing fours and eights. This is what gave the Al Green drums the sound of a "tom-like thing" on the snare. The drums there were not in a booth, but were pretty well surrounded by tall, dead baffles.
At the old Ardent, we had a drum riser-booth at first, about 2-3 feet off the floor, with "walls" built around it extending upwards about another 4-5 feet, the front wall having some plexiglass. We used this a while, but then took it down later, and put the drums in a corner of the room, usually with about 4 foot baffles around them. Console there was Spectrasonics, and machines again Scully, until we went 16 track, which was a 3M.
The drums, esp snare, were tuned pretty low, and were quite deadened. We almost always put the drummer's wallet on the snare drum. It would "jump up" when the drum was struck, providing a little "sound," and then fall right back down (gravity, I think!), deadening it again so that the ring was not very long at all. Rarely were toms hit. Almost everything was the bass drum, snare and hat (which wasn't mic'd).