King endgames can be tricky

White was wrong to simplify to the losing pawn endgame: 30 Red2? was a mistake and 30 fxg5 was OK.
Black advanced the wrong pawn 37...b5?? which allowed white to block 38 b4 so two pawns hold off 3 pawns. Black should have advanced his candidate passed pawn 37...c5 with a winning endgame.

I wonder in which country these types of endgames are called “King endgames”.

@Tenakel said in #3:

I wonder in which country these types of endgames are called “King endgames”.
Uh? I just said so cause it's shorter. Maybe K+P? But we surely know it's not two bare kings.

@Koalphin said in #4:

Uh? I just said so cause it's shorter. Maybe K+P? But we surely know it's not two bare kings.

Since you too should develop further, I'll give you a little tip:
When you get the chance, just look through an endgame book; there you'll find the names of the different endgame types. - However, your term will not appear anywhere.

@Koalphin said in #4:

Uh? I just said so cause it's shorter. Maybe K+P? But we surely know it's not two bare kings.

Yeah, that would be immediate draw.

But as kings are always on the board "King endgame" is rather meaningless.

"Pawn endgame" - and sometimes "King and pawn endgame" is used in those cases.

With more material on the board, usually only the pieces are named, like "Rook endgame" usually refers to anything with (single) rooks for each side plus optionally a number of pawns. In a "Rook vs pawn endgame" only one side would have a rook (optionally with pawn(s)). With two rooks per side, it's a "Double rook endgame" etc.

The tip to have a look at an endgame book is good. The table of contents will get you started on the right foot.