Why does that distinction need to be made?
Checkmate immidiately ends the game and that is in the rules. As is the rule that kings can not move into or through an attacked square. The idea that the king can't be captured flows logically from those rules.
In chess when a piece is captured it means that the captured piece is removed from that square and replaced by the attacking piece. This cannot be done with the King, therefore the king cannot be captured!
In Checkmate is the King removed from the square and replaced by attacking piece? No of course not. What really happens in Checkmate? The King is in Check with no way to get out, this does not mean a capture in chess terms, but does mean the end of the game. Therefore the King cannot be Captured!
Is there any book out there that says that the King CANNOT BE CAPTURED?
I have seen people play what I call their version of chess which includes taking out the King and continuing the game. Clearly they don't know the rules of the game. When I tell them that the King cannot be Captured in chess they always look at me in surprise.
Wikipedia does use the word Capture to explain the King piece. It says "the threat of Capture" a statement that could easily confuse people that don't know chess. If Wikipedia had used the words "The King Cannot be Captured " then it would make more sense to say "the threat of capture".