♡ 100 ( +1 | -1 ) definitely not possible(b)The king is said to be 'in check', if it is under attack by one or more of the opponent's pieces, even if such pieces cannot themselves move. Declaring a check is not obligatory. A player must not make a move which places or leaves his own king in check.
This is quoted from the "complete rules of chess" on GK site. In short and reduced to your question: Check is check, even if the Checking piece can't "capture" the King. The King is not allowed to be put on such a square "under attack". It is either not allowed to answer a check by pinning the checking piece so that it can't move. It doesn't need to move, the square is "under attack" - at least for the opponent's King.
In blitz games it is allowed to capture the opponent's King. Who first captures it is the winner. So in your example Black would move 1. - Bxe3 and win immediately. In normal games White's 1. Ke3 would be an illegal move and White must look for another one - first prioirity with his King because he touched it.
♡ 38 ( +1 | -1 ) New Laws of chessHi Misato
Note that the taking of the king will be illegal as of the 1st July. One must point out that an illegal move has been made. Also the illegal move cannot be claimed if a piece has been touched. Once a piece has been touched a move must be made with that piece.