The actual way to do quick, consecutive melee attacks.
There is another listing under weapons (titled 'Useful Melee Cancel Tricks') which says to hit the melee attack button (each button is different for each person depending on their controller settings), the reload button, and then the melee attack button again in order to double melee.
First of all, if you do this quickly your first attack does not connect at all; not to mention the fact that you can do this as many times as you want (not just twice), and the last melee is the only one that will do any damage at all. Secondly, if you wait until it does connect then you no longer have the option to melee cancel because the melee attack has already been completed. As a result, your second melee attack doesn't come any faster than pressing the melee attack button twice (and making that second attack come sooner is the point).
Here's the actual way to execute as many consecutive melee attacks as it desired at a slightly faster rate than your opponent: note: I use the Boxer control setup, which makes this trick slightly easier to pull off.
(1) Press the melee attack button, or L trigger using Boxer controls (you'll see why this is important soon).
(2) Immediately afterwards press the weapon switch button, or Y (this is why its easier to pull off using Boxer controls: its easier to hit the Y button immediately after the L trigger than it is to hit Y immediately after B). The specific timing of the button presses doesn't matter so much. As long as you do this part as fast as possible it will work.
(3) Wait for around 1/4th to 1/2 of a second, then press the melee attack button again. The timing of this is crucial. Otherwise the second melee attack comes no sooner, and in some cases can actually take longer to execute. The best idea is to begin by waiting for the instant you here the sound made by whatever gun you switched to (each gun makes a distinct sound when armed, the very second you hear the beginning of that sound is when to press the melee attack button again). After getting used to doing that you should be able to find a 'sweet spot' that makes the second melee attack come even more quickly.
*If you want to execute more than two melee attacks, or are in a situation where more than two are needed, its very simple to do. First, you go through each of the three steps described above. This executes the intitial two attacks. Then, you start the process over at step (2) as many times as necessary. The most consecutive hits I've achieved using this method so far is 11. I did this while hitting a wall and waiting for the sound of the melee attack connecting/the vibration of the controller that is caused by a connecting melee attack. Any opponent would be dead long before taking 11 melee attacks, even an overshielded one, but its good practice to see how many you can get in a row.