I'd have to say the endgame begins when endgame strategy comes into play.
When does the endgame begin?

There are 3 main groups of strategy: opening strategy, middlegame strategy, endgame strategy. Throughout the game certain positions allow for a certain percentage of each of the three groups of strategy. For example, the opening will have >70% opening strategy, <50% middlegame strategy, and <20% endgame strategy. Looking at those numbers, we know it's the opening. The same will apply with endgame.
Title says it all: when does the endgame begin and the middle game end? I’ve often used the syllogism “when one side has fifteen or less points* on the board,” but what are you’re thoughts?
*points being the generic queen=9, Rook=5, bishops/knights=3, and pawns=1.
If you include pawns in the definition that implies an endgame can lead into a middlegame.
Also your definition would mean that a lot of EGTBs should be called something other than EGTBs.

In general the endgame is when it's safe for the king to participate in the fight without fear of being checkmated, and so the main focus turns to queening a pawn while direct attack on the king is ignored as a possibility.
Title says it all: when does the endgame begin and the middle game end? I’ve often used the syllogism “when one side has fifteen or less points* on the board,” but what are you’re thoughts?
*points being the generic queen=9, Rook=5, bishops/knights=3, and pawns=1.