The difference, as I understand it, is that in Quake3 based games “crouch” brings your body down to meet your legs, whereas in Half-Life based games “crouch” brings your legs up to meet your body, which, if you’re not jumping at the time, is the same thing.
If you are jumping however, they’re different, in that the Half-Life jump-crouch will get you onto objects (boxes etc.) that you wouldn’t be able to get onto by jumping alone. I found this to be a bit weird when I first started playing RTCW after playing Half-Life.
As for strafe-jumping, Half-Life is based on a modified Quake2 engine I believe and there is no such thing as strafe-jumping. Whether that’s because of the Quake2 engine or the modifications they made to it, I don’t know.