Eh, I haven't understand a kanji of that website, but the pictures are sweet! Even if with this new graphic layout it looks more similar to VIRTUA ON rather than WARROID 
For 2 players VS games, I recommend 3 games I used to play a lot (not really sure if the names are correct since they were japanese).
Test Driver
Human Missile
Sum's Game
The first one is a racing game in a track that was a circle (or maybe an oval) the second one was like bomberman in the sense you had to kill the other player but with missiles instead of bombs, and you could control the missile, and the third one I don't remember very well, you had to go down some ropes and pick treasures trying to avoid crabs if I recall correctly.
They were all MSX2 games and featured the same character, a little kid who had always something pouring out of his nose 
If I will be very honest to you...
I am one, who has never ever even played F1-Spirit.
I never did. It just didn't appeal, I think. But I am not one to play racing games.
My little brother has played Hyper Rally, was extremely good at it, and I watched him play.
I can play Road Fighter a bit better than him - and that's all.
Though I must admit to have never really liked racing games - apart from Nyancle Racing, maybe. That one I love. 
I have fond memories of the "Le Mans" MSX game; sound and gfx wise, but never liked the game.
Also, a game like Greatest Driver... yeah... no. I love the introduction, but the rest, no.
I DO want to point you to an MSX game called "Lotus F3".
It's a racing game by dvik/Joyrex which pushes the MSX to its limits and is actually extremely fun to play.
If I will be very honest to you...
I am one, who has never ever even played F1-Spirit.
You may be punished because of that when the judgement day comes
Kralizec Tetris I would say. Me and my wife play it a lot and it's a real competition (if not war ).
The tank game mentioned earlier is probably Field Master.
Yes, that's the one! All written in MSX-basic