But when I used Define New Tags and created a tag called OriginalModDate2, it seemed to work in MB just like the hijacked one, but it doesn't show up when I look at the file properties in Explorer. Is that a limitation of how Explorer presents the properties? Or is one tag less "official" or standard than the others?
Yes, that is expected. Explorer will only show the list of properties it has been told to look for, which won't include any custom tags added in MusicBee. It's possible some other programs might read them but as a rule, you shouldn't expect any custom tags to be usable outside of MusicBee.
There are programs/add-ins that might let you work with the custom tags within Explorer if that's something you need to do.
(Just an example, not an endorsement - I don't use it myself)
It's not clear where "standard" tag functions leave off, and where MB features take over. For example, both my original hijacked tag and the one I made from scratch are listed as custom tags, numbers 1 and 2, respectively.
How much you want to dive into it this depends on your own pain threshold. I'm sure a lot of people spend years building up digital music collections and trying to work this all out and eventually just accept there's a point at which "magic happens".
If you want to read further, this is the closest thing to a list of "official" tags that you could reasonably expect most programs to at least be aware of, if not explicitly support in some way:https://id3.org/id3v2.3.0
Not all file formats will follow these same conventions, but because ID3 has been around the longest and was supported by MP3 before other formats (i.e. OGG/FLAC/M4A) even existed, the later formats should have an equivalent for these tags too.
PS. I know that none of this will let me actually update the file's mod time. But having a record of it that travels with the file is sufficient. And I'm thinking that somehow, if I have that info in an accessible tag, I'll eventually be able to turn up some method of transferring it to the mod time if if seems necessary.
Some nice person might write a MusicBee plug-in that could read your custom tag and use it to set the file modification time.
Bee excellent to each other...