Checking distance in K,R & P vs. K & R

After some embarrassing play at the Illinois Class, I have decided to take another run at getting basic endgame theory straight in my head.  Since I figure an expert should know Philidor and Lucena, I'm starting with King, Rook and Pawn vs. King and Rook.  My resources in this endeavor are Fundamental Chess Endings by Muller & Lamprecht, Batsford Chess Endings by Speelman, Tisdal & Wade, Comprehensive Chess Endings by Averbakh, Silman's Complete Endgame Course by some guy whose name I don't remember, and the 5-piece tablebase in ChessBase 8.

Here is a position from Batsford Chess Endings (p. 334) which I have looked at in the past.  1...Re1 is the only move that saves the draw for Black, and the reason given by BCE is "This move stops the e-pawn's advance after Ra8+ Kf7."

Here's the kind of position I might find myself playing as White in a tournament with my time running low.  I vaguely recall the position from BCE so I am somewhat encouraged when my opponent plays 1...Rg1+.  I'm not sure it's a blunder, but I'm pretty sure that 1...Rf1 would have drawn.

After 2.Kf6 Kg8 3.Ra8+ Kh7 4.Kf7, I am once again encouraged when my opponent plays 4...Rc1 because I know that checking distance is important and that his rook would be better off on the b-file.  So I play 5.f6 Rc7+ 6.Ke6 Rc6+ 7.Ke7 Rc7+ 8.Kd6.

Then my opponent suddenly scoots off with 8...Rb7! and I am left wondering whether I missed something or whether I never really had anything in the first place.  If I am lucky, I still hold the draw.  If not, my time runs out while I'm trying to figure out what happened.

So let's go back to the first position where 1...Re1 is necessary according to BCE because it "stops the e-pawn's advance."  This reason now strikes me as just plain wrong.  The tablebase tells me that the Black can draw even if the pawn advances to e7.  The reason 1...Re1 draws isn't because it stops the pawn from advancing.  The reason is that it forces White to bring his rook to e8 which allows the Black rook to take the a-file!  2.Ke6 Kf8 3.Ra8+ Kg7 4.Re8 Ra1!  1...Rd1+? loses to 2.Ke6 Kf8 3.Ra8+ Kg7 4.Ke7 Rb1 5.e6 Rb7+ 6.Kd6 Rb6+ 7. Kd7 Rb7+ 8.Kc6 when the Black rook can't scoot away.  BCE's comment that "the rook is misplaced on d1" isn't particularly instructive either.

The example in the second diagram actually comes from Fundamental Chess Endings rather than one of my games.  It gives 1...Rf1 as best but says. "1...Rg1+ is less accurate as White can penetrate further . . . although this still isn't sufficient to win."  I think that sentence would be much better if it ended with "because the Black rook still has adequate checking distance on the side."  It's not that BCE, FCE, and CCE don't mention checking distance frequently, it's that they don't cite it as the reason for a move when it plainly seems to be.

I would be very happy if anyone else would like share their experiences trying to learn these types of endings.  I feel like being able to articulate a better explanation for a move than I find in any of my books is itself a real step forward.  Of course, the test will come the next time I have to play one of these positions in a game.