I am Irish, so I presume I speak Hiberno-English! (and Munster Irish, rusty.)
I imagine that you would sign your name "on a birthday card" or "on the dotted line" or "on a cheque"?
If that's true, is there something about "the paper" that has different connotations?
I have used the phrase "put my name to (a contract or agreement)", but I've also "signed on to do something", while my boss has "signed off" (on) allowing me to do it.
Tá an Béarla chomh deacair sin!