Somebody should write a book about this theme! (if one hasn’t already been written) Every tournament one can find examples of players, even experienced players, stumbling at this choice. Remember this: it is always the other Rook!
Morriss, Pete (2032)
World Seniors Katerini 2014.10.31 Position before Black’s 31st move. The grandmaster played imprecisely a bit before, throwing away a large edge. Now White has dangerous play, threatening 32.Qh4 (winning on the spot!). Fortunately for Black, he has a defence based on White’s weak backrank (no luft!).
The question is which Rook should Black play to c8? After the game it was determined that Black can hold with 31…Rfc8! For example: 32.Qh4 Rc1+ 33.Rd1! RxR+ 34.BxR Kg7! 35.Nh5+! Kf8! (the whole point of moving the King Rook in the first place) 36.Qd4! and it appears that the game should be a draw, for example 36…Qd8 37.Qg7+ with a perpetual check.
INSTEAD, in the game Vasiukov moved the other (wrong!) Rook to c8:
This move denies Black the necessary square f8 for the King as in the previous analysis. Now White wins by direct attack.
32.Qh4 Rc1+ 33.Rd1 Rxd1+ 34.Bxd1 h5 35.Qg5
The only way to avoid mate is to give up the Queen.