I use pressure from the opposing side of the neck to bend strings. So if I am playing on the high B string and I want to bend a note, my thumb moves to the high side of the neck, so as to provide bending leverage to move that B string. At the same time I maintain the downward pressure, holding the string against the fret.

So I guess my answer is that I use both downward and sideways pressure against the fret board when I bend a note. I also double and triple my fingers pushing on the bending string to provide increased force to move it. I use the extra fingers to help push that bending motion. I also use those extra finger tips to mute unwanted string noise. So, it is a complicated process. After years of practice I am still working on this one, but you are on the right track. I would start with getting the bending motion to work for you and add the mutes as you gain bending proficency.