I just purchased a 75 mt125 for my girlfriend, been sorting through a bunch of problems, most are fixed now.
Your headlight and the blinkers are powered in different ways and should not be interacting the way you are demonstrating in the video. There are three coils on the stator as follows:
1) delivers ignition to the coil
2) delivers power to the headlight
3) charges battery (battery powers all other lights, including blinkers)
The stator is located behind the flywheel, left side of engine (where you adjust the points).
From your video it appears you have a wiring problem, as your are trying to run the headlamp off the battery, which it can't do.
Here is a link to an owners manual for your bike, including a wiring schematic. Shouldn't take too long to ID the problem with it: MT125 Owners Manual
There is also a silicone rectifier which is under the left side cover (the only cover) and mounted on the frame of the bike. I don't believe that this is related to you lighting issue, which appears all wiring related.