I watched your video 5 or 6 times. There are alot of possibilities and missing info here. Since you are overseas I'm assuming you are feeding the box with 240vac. Since I didn't see a rectifier I'm also assuming that everything in your control box is AC. Also, the transformer, since I saw a white and green wire, I'm assuming it's a 240/115vac step down, although it looks like it could be a 240/24vac stepdown. It also looks like the purpose of this whole thing is to regulate your gas valve to control mash/sparge/boil temps. IS the switch powering the transformer, or is the transformer powering the control box?
Big safety issue. That control box needs to be grounded ! ! ! !
Did you, or could you take a picture of any wiring diagram you drew and post that along with the specs of the controller, switch and light. Also the manufacturer and model numbers of the switch, light and controller.
Some of the possibilities are;
1) Your switch is wired incorrectly or not enough contacts on the switch.
2) As kyle187 pointed out, your light could be in series and dropping to much voltage so there isn't enough energy getting to the contactor for it to pull in. This is the most probable cause. After watching the video several more times, I think I have it. Take the hot jumper from the controller to one side of the switch. Take the hot to the light and the hot to the transformer from the other side of the switch. Then rewire the other side of the light to the common or neutral and it should work.
3)Also, if you are trying to pull in the contactor, without powering the controller, and the contactor is part of the controller, my guess is that it's not going to work.
4) You could also have a neutral/grounding issue. If the secondary of your transformer is single phase (115 or 24 vac) one of the poles on the secondary side should be grounded. You can test this by measuring voltage on each pole of the secondary side to ground. If you have half of the output voltage on each pole, then it is improperly grounded. If one pole reads 0vac and the other reads full voltage (115vac or 24vac) then that portion of the circuit is good and we are back to one of the first 3 possibilities.