Just "e6" would likely be enough context in most situations. Sometimes only one piece can legally move there. Sometimes it's obvious which piece should move there once the position is pointed out.
Now imagine what people like the Super GMs are capable of.
https://www.youtube.com/watch?v=eC1BAcOzHyY
They put a number of mid-game positions on the board, and Magnus was able to guess the players, tournament, game number, who won, what the next few moves were. Who was playing on the table next to him. What their moves were.
First, those people should look at his YouTube. Obviously he's capable of analyzing games. To think he'd be incapable of analyzing a game he just played? What? It makes no sense.
To think that someone, even if they were cheating in every game, was not a 2600ish rated player and to perform like Hans is just ridiculous. Every 2600 player can out analyze anyone who is a 2100 player (Botez).
The argument doesn't stand up at all.
They remember all the variations they consider, and they've considered most of the variations their opponents have calculated, so the variations aren't new branches, they're just pointers to spots in the game tree both players have in their heads.