I mean... I come from the forum scene and they were always called PMs, as in Private Message. What in the hell does DM stand for and when did things change from people calling them PMs to DMs.
Twitter used Direct Message instead of Private Message because they needed to establish that there is legally no understanding of privacy for DMs, because Twitter will surrender the contents of DMs to law enforcement / government / data collection. Fediverse should also use DM because there is again no guarantee of privacy (and you do not expect there to be).
Still, the conversation is private, as in not public. Even if it is surrendered to the authorities, the'll probably never be released publicly (unless leaked).
I suppose it's a slightly more accurate term. The messages here are not truly private since they are not encrypted, but since they are sent directly no one should read them in the normal course of using the platform. Calling them private might imply to people that other people cannot read them, rather than the reality that it is just very unlikely anyone will. I would also argue that if something is released to an authority it is not "private" even if it is not publicly available.
Honestly, it doesn't really matter which you use. People will generally understand either way, so you can go ahead and keep saying PM and others will say DM and we can all just understand that they mean the same thing.
I ran into a person the other day, I wrote "I'll PM you" and he replied "What is PM 🤨" 🤷. They may be young or just a late internet adopter, but still, that was just a sign for me that I should probably adopt the new DM lingo, as some people just don't understand what I'm talking about.