King, Rook and Rook's Pawn vs. King and Rook

Consider the following two positions which were inspired by Robert Moskwa's recent game with the World Under 8 Champion, Awonder Liang. White has just checked the Black king with his rook and Black has the choice of moving away from the White king and pawn with 1...Kd7 or towards them with 1...Kb6. The only difference is that the White rook is on c4 in the first and c3 in the second.  Both positions are theoretical draws if Black makes the correct choice. See if you can figure what the right move is in each case before you look at the analysis in the next post.