When writing music for piano, what is the most professional-looking way to notate that you should play the right hand notes, or the top staff, during only the second time through a repeated section? Do I write just "Play 2nd time only" above the top staff? Do I write "Play top staff 2nd time only" or "Play right hand 2nd time only?" Are there any precedent examples in professional repertoire?
If it's just one hand that's different and the remaining hand stays the same on the repeat, then your intuition is absolutely correct. It's very common for a piece to have an instruction that says "2nd time only" next to a staff. See also Octave changes on 2nd time
Similarly, you'll occasionally see notation like mp-f which means "play mp the first time and f the second time."
Now, if you wanted both hands to be different the second time around (or a portion of the second time around), you'd consider the volta. More colloquially known as a first and second ending, the volta changes the very end of the repeated section:
'Play R.H. 2ndX only' is fine. If it isn't a solo piano piece (where it's unlikely you would play NOTHING the first time round) make it especially clear that the L.H is to be played BOTH times. In fact, mark the lower stave 'play both Xs' anyway. Someone will think you want ONLY the R.H. second time.
If this is for a heavy-duty sight-reading session, write it out!
Don't worry about looking 'professional'. Worry about being clear. (Same thing, really.)
One option to consider is simply writing out the two repetitions. This especially makes sense if the right hand part that is introduced on the second repetition significantly changes the feel of the phrase. Writing out both repetitions like this would have a bit more of a connotation that the two phrases are significant developments of the theme as opposed to minor variations.