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?