Number of players

Mobygames has such info under "Specs" of a game.
www.mobygames.com/game/7301/mario-bros/specs/

This can be done with tags if we keep it simple like with the available options there:
1 Player
2 Players
1-2 Players
1-3 Players
2-3 Players
1-4 Players
2-4 Players
1-5 Players
2-5 Players
1-6 Players
2-6 Players
1-7 Players
1-8 Players
1-9 Players
1-10 Players
1-12 Players
1-16 Players
1-20 Players

I don't think tags are great way to go for this.

There's bunch of games that are "up to X" players with X being kinda arbitrary.

Bunch of games have the number of players limited based on game mode too.

Examples
- League of Legends IIRC has 3v3 or 5v5
- Bunch of boomershooters tend to have server-side configurable player limit that ranges up to 60 or so, some with no actual limit besides recommendation (Natural Selection 2 had default of 32 or 64, and I think this has been the case with Quake Arena games and those that mimic them)
- Bannerlord or some such recent game had default player limit of 250 or so even though it didn't have official servers (IIRC)
- Overwatch had 6v6 (and different mode with 3v3) and other similar multiplayer heroshooters have 12-ish players total.
- Vintage Story has no player limit but the servers apparently start having issues after 60 or so especially if everyone is active.
- Baldur's Gate is for 1 to 6 players and is designed for solo.
- Neverwinter Nights is designed to be played by up to 4 players, but the game on technical level supports up to around 60, and that isn't even hard limit.

There's also bunch of games where there's set number of player slots where the ones not taken by actual players are filled by AI, this is especially common in 4X games, like Stellaris and Civilization. These kinds of games also have the max number of players dependent on map size or similar.

Yes, I actually don't want to be too specific, like knowing the difference between 10 and 12 players.
I'd like to answer to these questions:

Can be played solo ?
We already have that, so the first number in the list above is not needed

If not solo, how is done?
Shared screen, lan, etc. we already have that

If not solo, how many players ?
Here the tags can be useful to state the max number of players, maybe rounding to power of 2:
2
3-4
5-8
9-16
more than 16 (and tag comment)

The special cases you listed can be solved with the above tags (even multiple of them) and a proper comment/article.

I'd personally like to see the following bits about this kind of info:
- Primarily solo (e.g. Baldur's Gate)
- Primarily co-op (e.g. Left 4 Dead)
- Primarily PvP (e.g. Planetside 2, Dead by Daylight, Quake Arena)
- Mixed PvP & Coop (e.g. Dead by Daylight, Planetside 2)
- Has squads (e.g. Planetside 2)

Tho I feel this is only tangentially related to the topic of this thread. Some of this info is already available via tags, probably.

Alternatively, rather than managing through tags, could a numeric field be added to the game editing screen for us to input the MAXIMUM number of human players - the other options such as single player, shared screen, split screen, LAN, hotseat etc, can be maintained and between the two we should be able to ascertain how the game is played and then add the necessary tags to expand on how it works with the various multiplayer tags (e.g. co-operative, deathmatch, last man standing etc.)?

Yes, this is another nice option and I like how it gets simplified message after message.