According to my book, for any subgame $sg$, $sg \neq$ the entire game, and also a subgame perfect nash equilibrium (spne) is a set of strategies that are nash equilibria in every subgame.
But .... since a subgame is never the entire game itself, this means that a spne is not even necessarily a nash equilibrium for the entire game. How does this make any sense? Should the definition of spne then not rather be that it's an equilibrium for every subgame .... + the entire game as well?
Naming the book may be useful.
I think what you are describing as a subgame, that is a subgame which is not the original game, is usually called a proper subgame. The definition of subgame perfect equilibrium is as you state: It is an equilibrium in all subgames, including the original game. You can check the definitions on wikipedia as well.
https://en.wikipedia.org/wiki/Subgame
https://en.wikipedia.org/wiki/Subgame_perfect_equilibrium