Stop looking at the files in Windows Explorer. Learn to use the Explorer function in MC, in the left Navigation Bar > Drives & Devices > Explorer. Just use that very carefully and learn how the Mode and Checkbox work. You can do a lot of damage in there, but it allows finding and checking file tags when the files appear to be "lost", or not in the Library. You can easily confirm if they are in the Library by changing the Mode.
Stop deleting files and re-importing them. I say that so often. It almost never fixes any problem with files in MC, especially when the problem is just the tags in the file. Learn how to use MC to find incorrect tags and fix them. It isn't that hard. Re-importing files just re-instates the problems that existed before.
Stop looking at Playlists to try to resolve issues. Sure, sometimes they are useful and the best way to do things. However, in your case here I believe that you need to learn to use Views, and to customise them to show what you need to see, because Views show the actual files that you have, while Playlists only show the files in the Playlist. So it would be easy to have a Playlist that you thought had a full Album in it, but it did not.
I have no idea why you posted an image of Playing Now in your first post. That isn't your Playlist. It is Playing Now, which is a special case of a Playlist, and is dynamic in MC. It isn't like a Playlist in iTunes or other applications. It
may well show the issue, but it certainly may not. It does show a certain lack of understanding of MC, I'm afraid.
Don't inspect Playlist files with mp3tag. Use a text editor, such as Notepad, or better yet, Notepad++. It is a great free text editor. Okay, I am a bit biased on this one, as I don't use mp3tag and do use Notepad++, so I don't know that mp3tag doesn't reformat or alter the files when viewed or saved. I could look I guess... not now though.
Understand that importing Playlist files can alter metadata (tags) for existing files in the Library. If the tags are correct, then you import a Playlist that has incorrect tags, your file tags can be updated to the incorrect values. For example, the "metal.m3u8" Playlist you posted in the zip files above includes the [Artist] and [Name] tags/fields. If they were wrong in that Playlist, then MC would be updated with the incorrect values. I haven't checked if anything was wrong in them. I leave that to you. But if those Playlists come from iTunes, check carefully, as they can have fields in the wrong order. The "metal.m3u8" Playlist does look to be formatted correctly though.
EDIT: Quick check. The Playlist file looks okay.
Note that there is no m3u8 extended Playlist export format in MC, and the m3u8 format does not include fields. If you want to compare apples with apples, and show fields, use the "M3U Playlist (extended)" format. That would have been another way to confirm the values of the [Name] and [Artist] fields in MC.
As BigSpider says, you have some tags wrong inside the files. Fix them in MC and all will be good.
But there could still be an issue with importing Playlist files. You could be completely correct in this statement:
The bug appears to be that, after importing the playlist, MC sometimes wrongly repeats the file name and the artist in the next entry. The rest of the fields appear correct. The filename is correct. So the correct song plays but it's just MC's database has partially wrong info.
I just can't tell from what you have posted, and neither could JRiver. I can't even see the full filename of the "E:\MUSIC\INBOX\1001 Albums\0873. System Of A Down - System O" example, where it does seem that the [Name] and [Artist] fields have been incorrectly updated, which a bad Playlist file import could do, as per above. But the "metal.m3u8" Playlist you provided doesn't have the error. You need to document exactly the steps you take to reproduce the problem, so that if it is a bug, JRiver can reproduce the problem and fix it.
I think this problem could occur if you accidentally selected two files in mp3tag when you updated the [Name] (also known as Title) and [Artist] fields. But you say you checked the tags in the file using mp3tag, so they should be correct. MC shows that they aren't anymore, in your image.
Also, terminology. Your image shows that the [Name] and [Artist] fields are repeated, not the file name and artist.