I chose to solder to the pins, as I wanted to put the pcb in a little box:

I hacked a PSOne DualShock pad, like you've got there, as it was the only pad that my ps2->xbox adapter would recognize..
If you flip your first posted picture, the pins are, top to bottom (no.16 is closest to the vibrators):
16: ground
15: square
14: X
13: O
12: triangle
11: r1
10: l1
09: r2
08: l2
07: select
06: start
05: up
04: right
03: down
02: left
01: analog - don't need it
I found it best to strip the flat white wire into 15 x 1cm pieces, then solder my colour coded wires to that, and then solder those to the pins.
Btw - see those connectors? Neogeo joypad extension cable. They are excellent, not much fatter than a PS controller cable, but contain 15 wires, so I can connect Snes & Segamega pads
