what means "if in range"? Does that just means that the King may capture an attacking piece? If the attacking piece is not 1-square away from the King I don't see how the K would have been able to capture it anyway.
You're right. The K may only move out of a check if it can capture its (singular) attacker.
If the attacking piece is more than 1 square away, it obviously can't.
and what means "capture out of double check". Here I understand that if 2 pieces are attacking the K, he cannot capture both of them. But that is trivial too, no?
It means that if the K is double checked, it is not allowed to move out of the double check. Not even by capturing one of the attackers.
So I guess that the important rule is that when a K is under check he can capture the attacking piece but it is forbidden to move the K out of check, to interpose a piece between the K and the attacking piece, or to take the attacking piece with another piece than the K. Is it correct? Or to say that when a K is under check, the only authorized move is if the K can capture the attacking piece.
I'll rewrite it to be more clear.
The important rule is that K can't move out of a check, except when it can capture its (singular) attacker. Other pieces are free to move, provided they're legal moves.
interpose a piece between the K and the attacking piece
This is legal.
take the attacking piece with another piece than the K
You're right. The K may only move out of a check if it can capture its (singular) attacker.
If the attacking piece is more than 1 square away, it obviously can't.
It means that if the K is double checked, it is not allowed to move out of the double check. Not even by capturing one of the attackers.
I'll rewrite it to be more clear.
The important rule is that K can't move out of a check, except when it can capture its (singular) attacker. Other pieces are free to move, provided they're legal moves.
This is legal.
Also legal.