*** Not a teacher ***
I think there is a lot of room for error and misunderstandings in these usages.
I think most people think "three times more" and "three times as many" are synonymous. Personally, I always use "three times as many" in an attempt to avoid confusion.
Tom's 20. Ben's 3 times older than Tom --> The vast majority of English speakers will say Ben's 60.
Tom's 20. Ben's 3 times as old as Tom ---> Ben is 60 - no room for confusion.
I think you are asking for trouble when you say "three times fewer". I would say it means:
Tom made 12 mistakes a day last year. This year he's made 3 times fewer mistakes --> (12/3) = 4 mistakes. But I can see how you could interpret it to get your equation (12 - 12/3) = 8 mistakes
It is much more clear to say "one third as many":
Tom made 12 mistakes a day last year. This year he's made one third as many --> (12/3) = 4 mistakes.