That isn't how I remember the dialogue, though I'll defer to what you say as you say you've seen it recently. Is it available on DVD?
Yes! And the supplemental material includes a documentary that they shot when the entire crew and all three principals went back to the mountain for shooting. Joe, the guy who fell, talks about how he's come to peace with the whole thing, but he starts to get weirded out by being there. Simon, the guy who cut the rope, keeps denying that any of it effected him--this was just one climb of many, etc.--but it's clear that he's not real happy about being known forever as, well, "The Guy Who Cut the Rope." It's a fascinating documentary and basically stands as the next 45 minutes or so of the original movie.
The other thing I didn't realize was that Joe played himself in some of the long shots of crawling off that glacier and through that moraine. Being forced (or at least contractually obliged) to repeat that is at least part of why Joe stops being so happy to be back.