As an American lawyer, that's an easy one to answer. "On cross" is short for "on cross examination." A witness is called to the stand to testify. When the lawyer who called the witness is done asking his questions (which we call "on direct" examination or just "on direct"), the opposing side's lawyer gets a chance to ask his own questions. This is called "cross-examination." So whatever the witness said in reply to cross-examination is what we say he said "on cross." . That's all it means. Then there's "redirect" and then "re-cross" if the back and forth questioning goes further.

I guess it means "on cross-examination". I've never seen the phrase in this short form before.

"on cross" is not a term I've ever heard of. Do you mean "on oath"?