If I recall, the "acid" part of "acid jazz" was almost tongue-in-cheek.
The people in the dance music world [house, techno, etc.] are notorious for naming all of these genres for music with only a little difference between. The prime example being the jungle craze in the 90s-you had two-step, jazz-step, hard-step, tech-step, etc., etc.
When the DJs were spinning these old soul jazz sides, they called it acid jazz. Just like the jungle thing, house music DJs were naming like crazy. One of the terms just happened to be ACID HOUSE.
I think...