I have asked Alex as this would involve the other sites as well. Please allow a little time for him to reply.
The mariokart64.com URL is the actual original URL, the mariokartplayers.com one was created so there isnt confusion with the game being associated with Mario Kart 64 each time. I always used this too.
EDIT: Update from Alex..
The easiest way around it is to use the existing "Comment" field per time which usually stores the driver name. Alex can turn the field into a link if it "looks like a link". This would lose the Driver name value, but if a person is interested in seeing the driver then they can just click the video as it would show anyway.
Alex created a working example here:
http://www.mariokart64.com/smk/profile.php?pid=12He set Armin's VL1 lap to have a comment that is a twitch url. The code for displaying the profile noticed that it's a twitch link and turn it into a link right next to the time. If I enter a comment that does not look like a link, such as "Bowser" again, it should be treated as before and just appear in the hover pop-up.
Please let me know what you think, and I will get Alex to use this implementation after I am back a week later if happy.