Yes, a "tune" is equal measure rhythm and pitch. After all, you can recognize the song "Jingle Bells" by a clapped out rhythm alone.

But if you said: "He sang out of tune" to a person raised in the Western 20th Century, he or she would likely think of a song with sour notes, first, rather than getting the timing wrong, as in the lyrics to that Lenon/McCartney song "A Little Help From My Friends":

What would you do if I sang out of tune,
would you get up and walk out on me?
So lend me your ears,
and I'll sing you a song
And I'll try not to sing out of key.

And also, you can find evidence of this switch in the way our American National Anthem is sung: notes are often held for so long that you can't comprehend what the original word was meant to be, but people still applaud louder the clearer and more pure the notes are (thus encouraging the singer to hold them for even longer).

I guess what I'm really asking is: Have our basic assumptions about which "comes first" in a tune really changed (i.e. that Shakespeare wouldn't have fliched quite as much over a wrong note as we would)? Or, does singing out of time create wrong notes, from the listener's point of view (I guess this last question is a technical one, based on how multi-part harmony works with two or more singers)?