Okay, let me try to asciigraph a little. (Ignore the periods; they're just there to keep things aligned. I hope.) In Type II, we have two strands, A and B, arranged like this:
A__..__B
.....\/
...../\
.....\/
A__/\__B

where strand A passes below strand B both times. A Type II move allows us to change that to this:
A__......__B
.....\..../
......\../
....../..\
A__/....\__B

or vice versa.

As for Type III, there are three strands involved:
B..........C
..\......./
A_\___/__A
.....\../
......\/
....../\
.....C..B

and A passes below B and C both times. Then we can slide A down below the B-C crossing:
.....B..C
......\/
....../\
...../..\
A_/ __ \ _A
../.......\
B..........C

A still passes below B and C, and we don't change the B-C crossing - whichever was on top before stays on top.

I hope that's clearer - both that the asciigraphs don't get messed up, and that they do clarify!