@TwoMove, right, he said everything but I needed to rephrase everything and metabolize it, so I followed here my path to get it. Maybe it will help somebody, or maybe it will help only me next time I will have doubt in a rook endgame, and I will remember that I wrote in the forum about it!
@RedMountainFox, to be honest I am so bad at calculating in endgames than having some easy rule to follow in some special case seems a big help to me. Not to mention, this was one of the drills you linked to me.
For me time is always an issue, I very often run out of time in 60 minute games.
But you stressed indeed an important point, that I already mentioned around here.
Classes, drills, and so on and so forth, very often focus on material that is too difficult. Maybe what we beginners should focus on is really little things and understand a few basic concepts, yet... I am decent at theory and then play terribly.
There are for sure some bad habits of mine that I should take out of the equation to improve.
50kg said all you need to know in the second post. Ke3 isn't winning because Ra3ch. King safety is not relevant in this endgame. In rook endgames, it nearly always a good idea for the weaker side to avoid exchanging rooks.