Basically this ending (pronounced Lou-Chen-Ah) is a staircase procedure. As you mentioned, you would think the height or vertical distance from Rook to King would be the governing factor. But Lucena themes are numerous, and in one of them there is a sneaky "end around run" which the width of the board allows.

It is a draw on an 8x8 board. Move it to a 10x8 board, it is a win.

First, a typical Lucena win. White king on e8, white pawn on e7, white Rook on f1, black King on g7, black Rook on d2. This is on an 8x8 board.

Lucena published a manual in 1497 (a mere 5 years after America was ‘discovered’ by Christoval Colon, a.k.a. Christopher Columbus) wherein certain endgame techniques were mentioned, but, curiously, the critical position was not in that tome! In truth this endgame is an age-old one, but an Italian author named Salvio was the first to document the winning technique for White in the year 1634, and even he attributed the composition to a previous mentor named Scipione Genovino. That historical tidbit aside, White can win independent of the side to move by using the “staircase” technique.

1. Rg1+ Kh7 2. Rg4! (placing the Rook far enough away from the enemy King to avoid a trivial banishment while allowing for the quickest check-intervention block as the King marches towards the Rook. Playing 2. Kf7 at once achieves no gain since the Black Rook delivers a long sequence of harassing checks.) 2...Rd1 3. Kf7 Rf1+ 4. Ke6 Re1+ 5. Kf6 Rf1+ 6. Ke5 Re1+ 7. Re4 and the win is trivial now.

Let's look at another theme.

White king on h6, white pawn on g6, white Rook on a7, black Rook on b8, black King on g8.

This position represents another common 8x8 endgame where White’s g-pawn is functionally useless. It appears Black’s Rook is serving a purely passive role, but a draw is all White can hope for with 1. Rg7+ Kh8! 2. Rh7+ and now Kg8 draws. Black needs to avoid only 1...Kf8?? which loses to 2. Kh7!! Rb1 3. Rf7+ Ke8 and now the pawn promotes.
Transplanting the drawn position onto the Gothic chess board results in a win.

Black loses the Rook at once after 1. Ri7!! since mate is threatened immediately with Ri8# if the Rook moves. After 1...Kh8 2. g7+ Kg8 3. Ri8+ Kf7 4. Rxb8 Black is toast. Black likewise does not have the resource 1...Kf8 2. Ri8+ Ke7 3. Rxb8 which loses even quicker.

I hope this analysis is accurate, I am doing this without sight of a board.