Been told many times that walljumping with W is wrong, and you see all them new jumpers try to do so. Might as well call jump1 scrubjumping. Or lazyjumps.
I think (hope) people only say that because new jumpers tend to try to airstrafe with W and it doesn't work. If you understand how airstrafing works, then there is nothing any different about W (or S) then A or D. It can make certain jumps easier, like the ones demonstrated in the first video. I myself use a variant of that for long left wall jumps. I look about 135 degrees back, hold W and D, and hold M1. It's not perfect (you need to look further forwards when you start going fast), but it's very easy and works pretty well for most of these jumps. A lot easier than trying to flick 135 degrees back and forth while alternating A and D.
Pro tip for left wall jumps: the way I do it is (using the default rocket launcher) start with a normal jump at about a 45 degree angle to get some height and some forward momentum, then just looks straight at the wall, a little bit up from straight down, and hold w and m1. Actually works surprisingly well. I can pass the long wall on raiin's BRAINFARTONMAPNAME_2 (Edit - Ayers2, shortly before end of detail), the sort of hell looking hall, every time down the left side using this method.