Was it really a lot, though? I feel like this something I've heard people who played 1E in the US say, but I was in the UK, where the Satanic Panic wasn't really "a thing" (outside of a few poorly-regarded tabloids and the like), and none of the many 1E grognards I played with believed this. The had a lot of mean things to say about 2E, but literally the only "oh that's because of the Satanic Panic" thing any of them ever pointed out was the re-naming of the Demons/Devils/Daemons. Which with respect were dumb names anyway, because those aren't even different things! That's like having classes called Fighter, Warrior, and Soldier!

Or Cleric, Priest, and Monk (oh wait...

).
What other stuff was there? I mean, 2E definitely has a massively more upbeat and less goth-y/metal-y tone than 1E, but was that "Satanic Panic", or was that because they wanted a broader appeal and/or D&D had changed?
Besides the Demons and Devils being renamed. . .
The very idea of Clerics worshiping fictional Gods was treated as an optional rule. By 2e PHB standard, clerics just worshiped the vague concept of "Good" or "Evil", and having fictional religions or deities for a game was officially an optional rule in the PHB. . .because a big part of the Satanic Panic was to say that worshiping any deity other than the Abrahamic one was an act of Satanism, and the theological idea pretending to commit a sin or contemplating a sin is as serious as actually performing the act, so pretending to play a character who worshiped another deity is the moral equivalent of worshiping another deity, and hence Satanism (or at least according to the logic of the religious fundamentalists).
Don't forget that Assassins were removed from the core rules too, that having an explicit killer-for-hire as a character archetype was something that was WAY too easily misunderstood. Also, removing Half-Orcs as a race from the core rules, because their typical or implied backstory involved rape.
Oh, and the "Satanic Panic" was VERY much a real thing. I could tell several stories of encountering it in the 1990's, but the short version of those stories:
1. When I first expressed interest in D&D, my father freaked out because he'd heard at Church that D&D was "satanic", and it took a LOT of time to convince him otherwise. He didn't stop thinking D&D was satanic until Hasbro bought out WotC, because I got him to admit that there was no way Hasbro, a publicly traded multi-billion dollar company who lives on a wholesome reputation, would put out anything so controversial or questionable.
2. When I tried to get friends in Junior High (in the early 1990's) to play D&D with me, it lead to rumors about me being a "devil worshipper", I discovered that some people thought gaming miniatures were actually religious idols (they couldn't wrap their mind around the idea of miniatures for wargame/tactical purposes, they assumed little metal figurines of fantasy creatures HAD to be for some occult worship purpose).
3. Those rumors got me a trip to the guidance counselor, because they had mutated into rumors that I was trying to recruit classmates for some suicide pact or satanic cult, just by asking some friends if they'd like to learn to play D&D. Just that question alone got people thinking "satanic cult" and "suicide" and reporting me to the authorities.
4. When I DID actually manage to start playing several years later, in college, I was still living at home and going to Church at a Church in my hometown. I was helping out at the Church one Saturday when I was leaving to go to my weekly game. The Pastor asked why I was leaving, so I was honest. . .and when he heard I played D&D, he basically kicked me out of the Church and said that I wasn't welcome to come back unless I burned all my D&D books and publicly repented being a "satanist", and that wasn't going to let any satanic cultists infiltrate his congregation. I left and never returned.