when does the endgame start?

Endgames are usually when there are few pieces on the board, often when the major pieces have been traded off, though that isn't necessary. There really isn't a hard and fast definition.

Regarding what constitutes a brilliant move here, I don't believe any staff have ever said. 

That is subjective, but when the king goes from defense to offense is a pretty good indicator that the end is near.

If most of the pieces are on the board still, it's not an endgame.

I'll copy paste my answer from here:
https://www.chess.com/forum/view/general/when-does-middlegame-switch-to-endgame?

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There's no quick and easy definition that works for all cases. In general it's when there's not enough pieces to threaten a direct checkmate, so the game takes on a strategic character where players maneuver to advance their pawn majority and/or target weak enemy pawns. Often the kings are safe to come to the middle and participate in the fight.

As a few counter examples, you'll hear people say "the Berlin endgame" as early as move 9 when it's definitely not safe to bring the kings out yet, and also a Q+R vs Q+R endgame you'll still want your king tucked away behind pawns.

So some positions linger between a late middlegame and early endgame... but yeah, I'd say it's the endgame when both players are more concerned with eventually queening a pawn than with any ideas of checkmate.

Whenever there are about 3-4 pieces (Rooks/Bishops/Knights) Normally Queens are traded or everything is traded but queens.

this is how Mednis defines it in "from middle game into endgame" of course the big caveat its subjective. 

When the middle game is over.

When the middlegame ends. 

I hope this helps, good luck.

Dagnabbit elio, u beat me to it. I hadn't even read any of the responses, it was just my first thought when seeing the title.

This is when the endgame starts. It's the point you reach in the game where you want to use however many pieces you have to win the game. This is how all endgames start.

Wow, who is this author?

"Central factors are not of particular significance"

One sentence later:

"The king should be centralized and utilized as a potentially valuable attacker"

Plus there an enormous number of middlegames that meet all 3 standards... very poorly written.

Oh, I see, Edmar Mednis.

Hard to understand how he could write something like that.