Absolutely. I think we should stop using PM and DM terminology as if they’re different things, and definitely shouldn’t codify the terms anyway in the UI, it’s an uphill battle for meaning we won’t win. The words behind the acronyms (private message and direct message) don’t differentiate a “type” in any meaningful way.
Wikipedia equates the two to mean the same thing, Merriam-Webster defines a DM as a private message… this just creates baggage for us to explain away.
Personal Chat and Personal Message seem like the clearest options to me. In documentation we might call them that, and colloquially it’s easier to say “send me a [chat] [PM]” and know the difference between the two.
The distinction between the two types of functionality is rather thin from my perspective, and probably non-existent for most users. Encryption is important for some sites, but how many? probably fewer than 10% of Discourse sites use it, right? Search is a factor for more sites, but I’d be surprised if we didn’t have it within the next year. There’s not much to stop me from writing paragraphs in chat (and I frequently do!) and nothing stops people from sending me chats asynchronously.
Reddit is one platform that I can think of that has the two concepts co-exist in a similar way, and AFAIK they’re gradually moving to all-chat for DMs. And people ask the same question there (their admin didn’t have a great reason for it either): https://www.reddit.com/r/help/comments/pali45/whats_the_difference_between_chat_and_message_in/
Anyway, we have reasons now… but they’re going to go away. When they do I think we should strongly consider what merging the two concepts might look like. It will make things much clearer for users in the end.