If you're at school writing something down and and someone bumps into your shoulder and made you mispell something as a result, is it right to tell at the person who bumped into you "you made me make a typo"? (By writing down, I mean using a pen to mark a piece of paper)
No. A typo is a typographical error:
American Heritage Dictionary "typographical error"
"A mistake in printing, typesetting, or typing, especially one caused by striking an incorrect key on a keyboard."
A typo is an error in spelling, that is, an incorrect letter or symbol.
Being bumped by another person when you are writing can't make you produce an incorrect letter. It will probably result in a smear or scrawl.
(The OP asked specifically about "writing something down". That phrase means to make marks on paper with a pen or pencil, and doesn't apply to any sort of typing.)
I wouldn't call it a "typo".
When I write "corect" instead of "correct" because I'm careless (I know how to spell it correctly but didn't) that's a careless mistake. When I write "corect" instead of "correct" because I think the word has only one "r", that is a spelling error. When I write "cprrect" instead of "correct" because my fingers hit the wrong key on the keyboard, that's a typo. When I write "corrct" instead of "correct" because my keyboard is broken and doesn't let me type the letter "e", that's a keyboard malfunction.
If I hit the wrong keys because someone bumps into me, and the wrong letters appear on the screen or on paper, I wouldn't call that a typo. I "typo" is a mistake by me, hitting the wrong keys. Someone bumping into me can result in messed up text, but it doesn't cause me to make a mistake - a mistake would be my fault, and this isn't.
I would say "you messed up my writing". There are probably many other ways to express this. Importantly, you want to put the blame onto the right person.
BTW. When writing on paper with a pen or pencil, you can't make "typos". A typo is today a mistake hitting the wrong key on the keyboard; in previous times it was making mistakes during typesetting.