What is the etiquette (or tournament rule) for the act of capturing a piece? There are a few different possible ways of capturing an opponent's piece. You could:


Pick up both your piece and their piece simultaneously with both hands, then set your piece down where their piece was.
Pick up both your piece and their piece using the same hand, and then set yours down.
Remove their piece from the board first, and then use the same hand to move your piece.
Pick up your piece, use it to shove theirs out of the way, set down your piece, and then remove their piece.


And perhaps there are a few other ways as well.

Which of these ways are the best etiquette? (The "shove theirs out of the way" method seems particularly inappropriate.) Are any of these ways illegal under tournament rules? (Don't the USCF rules say that a move must be made with one hand?)
Seeing how Super Grandmasters capture pieces can be instructive:
Garry Kasparov
Capturing an adjacent piece:

Capturing a distant piece:

Source: YouTube
Carlsen (white) vs Caruana

Source: YouTube
Aronian (white) vs Morozevich

Source: YouTube
Hikaru Nakamura (white) vs Vladimir Kramnik

Source: YouTube
Summary:
With the exception of Kramnik, captures proceed as follows:

Lift the opponent's piece and palm it.
With the same hand, move your piece.
Stop the clock using the hand holding the captured piece.
Table the piece.

It's worth noting that it's OK to table the piece in your opponent's time; you do not have to table it before you stop the clock.
Kramnik (playing black in the final image) lifts his own piece before transposing it with Nakamura's pawn in one quick snap. Kasparov also does this with the pawn capture in the first gif. This isn't wrong, it's just a different – and perhaps more aggressive – style. It does not matter if you lift your piece or your opponent's piece first, but all high-level players touch both pieces and the clock with the same hand. (Note that Magnus as white tables the piece on his right even though the clock is on his left.)
It's probably safer not to use two hands for captures or castling, as tournament rules vary on the use of two hands, and it's been a source of controversy in the past. e.g. Giri vs Epishin, Supercube Blitz Utrect, 2009 (scroll down to &quot;The Incident&quot;).