...why are we changing this yet again? I thought "specific people" was straightforward and to-the-point enough...

The existing web UX has "Private mentions", makes sense to match that wording...

If it were me I would rename the so-called "Private mentions" view, since a lot of people find that term extremely confusing: #24837 (37 thumbs up, in the top 100 issues; n-way tie for 79/100)

The "Specific people" label on the compose box makes (relatively more) sense, and the view of such statuses/conversations could maybe be called "Conversations". (It's literally /conversations.) But it would be more productive to eventually consider an actual conversation system that isn't just limited to ones involving direct statuses.

Broadly speaking, conversations can be bound loosely by a reply tree (the current paradigm), or they can be bound by an explicit context that is owned/moderated by someone (more forum-like or topic/thread stuff), or they can be bound by a set of participants or audience (more chat-like or room/channel stuff). The UX confusion comes from people seeing the 1st thing (direct statuses existing in multiple separate reply trees) and assuming it works more like the 3rd thing ("DM"; messaging a specific mutable audience of 1+ people), because they are more used to the 3rd thing being the prevailing paradigm. Perennially renaming the one feature we have is just going to cause churn as people don't actually gain a better understanding of the feature, they just lose whatever familiarity they have with the old term.