A very interesting question about chess rules. Checkmate unavailable?

Hello everybody! I am a chess beginner, and im quite familiar with general game rules, regarding when a game is won/drawn (when a king is checked, and has no legal squares to move, it's a checkmate, or if there are no moves available at all, it's stalemate, which means a draw). but recently i was playing a game with a friend of mine, and after a terrible mistake of mine, we got into a ridiculous position: black queen moved to f6, obviously checkmating my king, cause all the escape squares are either blocked by white pieces, or attacked by black. But there is a little nuance: why cant white king go to e3 square? It is covered by the knight on c4, but the knight couldn't possibly move and "take" the king, cause it is pinned to the black king! in fact if the knight moves, it would be checkmate for black. So it is kind of "double checkmate" in this situation. My friend didn't know how to assess this position either, although she is a much better chess player than i am. The question is: can the white king legally move to the e3 square, and if no, why not? It can't be legaly captured by the black knight anyway. In my understanding, "checkmate" means that the king could be taken on the next move, but in this situation on e3 it couldn't be taken, cause of the pin. Please clarify this if you can :)

I see it this way: your King can't go to a square which is covered by an enemy piece, regardless if that piece could "take" or not

But the e3 square isn't really covered, cause the knight is pinned. It can't legally move

Here is the thing about "pinned" pieces.  The game ends immediately when the king is captured.  Whoever captures the king first wins.  Therefore, a pinned piece may move to capture the king because it will capture the opponent's king before the king it is pinned to is captured.  The game ends instantly when the king is captured; it doesn't matter that the pinned piece will give up its king on the next move.

if your king is taken first you lose automatically regardless even if you take his king after he takes your king

Ok that's clear, but when i play online f.e. it doesn't even let me move a piece that is pinned to the king, cause it's not a legal move. Even in tutorial videos they say if a square is protected by a pinned piece, it means it's protected 0 times 

You guys need to get a rule book .  The diagram position is checkmate and  white loses .  

This amounts to the argument:

I can move my king into an attack (Ke3) because you can't move your king into an attack (knight is pinned).

But this is inconsistent. It either applies to both players, or neither player.

As others have this, this is mate. If it's easier for you, think of it as whoever's king is captured first loses.

Think of it this way.

Once the Knight takes the White King, the game is over. 

Once the game is over, it no longer matters what may happen next.

Although it is a neat trick to think in terms of which King is "captured" first, in chess the King is not captured.

So you just need to think this way: a pawn or piece that attacks or controls a square does not mysteriously lose those powers simply because it is pinned. It may not be able to move, but it does not lose its ability to attack/control a square. So a pinned piece is perfectly capable of delivering mate or supporting another piece that delivers mate. In this example the knight is controlling a flight square. It is irrelevant that it is pinned.

A pinned piece cannot give checkmate, because the pinning piece could capture the pinned piece. Unless the pinning piece is also pinned... but whatever. But a pinned piece can give check, and do other stuff.

That is such a beautiful position. How does that happen in a real game?

Anyway, all the people saying it's mate are right. That's the official rule, and it does make sense because ordinarily you can't move a pinned piece because your king would die, but if you're moving it to kill the other king it doesn't matter because his dies first.

A king can't move to a square that is being attacked notwithstanding the attacking piece is being pinned. You can't argue an illegal move is legal because it is illegal for another piece to move. The basis of both moves being illegal is rationalized on the same principle, that the king will be captured next move, and a king can never be allowed to be captured voluntarily. White can't move to E3 because it violates that principle, we don't have to discuss blacks next possible move because white can't make the move without breaking the rule that you can't move your king to an attacked square. If you violate that rule, black has every right to violate it too and will capture the king next move with the knight. Game over.

A pinned piece can give checkmate where the pinning piece is itself pinned.

Or where the pinned piece delivers discovered check as part of a double check.

There is no such thing as "taking the King". King cannot be taken. If King is in check and cannot move it's a mate.

There are many debates when beginners playing with physical board without director. "Hey, what's that move was?" "it's en passant". "No, that's illegal." "NO".

And rules about castling:

"You cannot castle because my bishop is attaking your rook." "Yes i can."

I suggest that you play at first online chess, where you don't have to worry about illegal moves. Sting in the tail is you cannot play 3D board on free sites if you are accustomed to play 3D board.

thanks everybody for responses! It's kind of a noob-ish question, but i hope you see how it made sense from my perspective. The position occured when i started a king hunt, and chased her kind all the way to b3, capturing all her pawns during it. I was about to give checkmate, and i was sure the e3 is a safe flight square for my king, cause of the pin. But apparently it wasnt :(