I don't get the definition of subgame perfect Nash equilbrium. A subgame does not include the entire game which begins at the very first choice made by some player. A subgame perfect nash equilibrium is a nash equilibrium in every subgame. Since we just said that a subgame never equals the entire game, this implies that a subgame perfect nash equilibrium is not even a nash equilibrium for the entire game .... what??!
Your trouble of understanding is caused by the assumption in the second sentence:
A subgame does not include the entire game which begins at the very first choice made by some player.
This is not true. A subset of a game can be any subset of nodes that (1) starts with a single initial node, (2) includes all subsequent nodes and (3) makes sure that the information set of any node in the subgame are completely included in the subgame.
Since the actual game does meet all three conditions it is a subgame of itself.
This being clarified, the definition of the Subgame Perfect Equilibrium (SPE) does make sense. An SPE must be a Nash Equilibrium (NE) in any subgame, including the whole game.
Note, that it then logically follows that every SPE is a NE (but not vice versa).
(Also, for further reference: The set of subgames excluding the complete is called the set of proper subgames. Note, that the definition of the SPE only refers to subgames without specifying that they have to be proper ones.)