I am roughly 1400. Of the 15 OTB games I played last year, two ended up with rook endgames - that's 13%. It is often said that at my level I should focus on other things, but whenever I encounter such a rook endgame I feel very lost:
The first position was
[fen "6R1/8/5r2/2K5/1P4p1/6k1/8/8 b - - 0 52"]
I was white, my opponent offered a draw and I took it.
Another one was
[fen "8/6R1/3kpp2/6p1/8/4PP2/r4K1P/8 w - - 0 36"]
Here I was black, my opponent offered a draw and I took it. In these positions, both my opponents and I were absolutely clueless on how to approach these positions, how to at least try for something. We were both afraid of making mistakes and loosing in the end.
This prompted me to start studying rook endgames more seriously. I've started with Lucena, Philidor, and also https://chess-endgame-trainer.web.app/ , with simple things like rook vs pawn. My plan was to try and practice over and over to really understand how to play these positions by heart.
However the first training position in that app for rook vs pawn is
[fen "8/8/8/8/8/p7/k1K5/2R5 w - - 0 1"]
where apparently Kd3 draws and Kd2 wins. I have no clue why and can't understand the plan/approach at all. I've also looked at Dvoretskys endgame manual for rook vs pawn, and couldn't find any theme that matches. Or I wasn't able to see the connection between the presented techniques and the position above.
Long story short: I feel there is a huge gap between theoretical rook engames (i.e. rook vs pawn or rook + pawn vs rook, like Lucena and Philidor) and practical endgames like the first two ones above. I feel that it is impossible to calculate such positions, and there are so many moves where there are three options which draw/win/loose but I cannot figure out why - like Kd2 vs Kd3.
So what is the best way to study rook endgames from a practical club player perspective? How to find plans or "test" your opponent in the first two positions above? Are there any guidlines like, i.e. in the second position: Is it better to check from the side or put the rook behind the pawns? Is it better to push the pawns and/or exchange them, or be very cautious? What is a good study plan for these things?