What is the correct hand movement to capture a piece? I see players in international tournaments taking pieces in the following fashion: first they take off the board the piece they intend to take, and only then, with the same hand, they move their taking piece in the square occupied by the removed piece (except in en passant, of course).

I was taught to take pieces using a single hand movement instead: I move my taking piece to the square of the piece to be taken and remove that in the same time putting it off the board.

Since I think that the first method is more time consuming, I am assuming that it is the only legal method to take pieces, while mine must be illegal. Am I wrong? Or does it depend on the particular policy of each tournament?
According to the FIDE Laws of Chess:


  4.1 Each move must be played with one hand only.


In other words, the "correct" hand movement is one which uses only one hand. The order in which the opponent's piece is removed and the player's own piece moved to the capturing square is irrelevant.

Note that once the captured piece has left the board it doesn't matter which hand is used to place it on the side. It is perfectly legal, for instance, to lift the piece from the board with the right hand, immediately transfer it to the left hand and carry it to the side of the board with the left hand whilst simultaneously moving the capturing piece to the capture square with the right hand.

Something similar applies with promotion. The promoted piece must be placed on the board with the same hand that removes the pawn from the board but it is perfectly legal to pick up a queen, say, with one hand whilst removing the pawn with the other and to transfer the queen from one hand to the other before placing it on the promotion square.