Your pics aren't the best, but here's my take:
In your 2nd pic, I believe the white wire coming into the green terminal on the right is +12VDC. It appears that there's then a jumper to provide +12VDC to the switch on the left via the blue terminal. The brown wire at the other end of the switches seems to be similar, i.e., it feeds into one switch, with a jumper over to the other switch. Not sure why this would be, unless it's somehow supporting the lighting of the switches, i.e., it's supplying the -12VDC side of the switch lamps. That leaves the middle terminals (looks like white wire on one and yellow w/white on the other) that would go to the LED strips. The -12VDC of the LED strips should just tie in to ground somewhere; it doesn't need to come back to the switches.
Suggestion: Pull the green terminal with the white wires on the right switch. If nothing works, I would suspect that proves it's +12VDC to power everything. Put it back on and pull the green terminal with the brown wires on the right switch. If the LEDs still work, but the switches don't light up, then that would prove this is the negative side of the switch lights. Now pull the center terminals one at a time and if first one LED strip doesn't work and then the other one doesn't work (but the switch still lights up), that should prove the center terminals feed the LED's.
Note that I could be backwards about the white and brown wires, in that the brown wire could be the +12VDC and the white wire could be the -12VDC.
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