Soldered it all with a single-thread wire (blue -
because I didn't have black).
Multiple-thread wire is poorly suited - it breaks off at the place of
soldering.
Everything is winded to the handlebar with a thin (30AWG) black wire.
Connectors should be covered with some putty - I had glitches under rain. Button's contacts are spaced far from each other, so should be OK.
Brake light buttons are glued (with "Moment") to the brake lever housing
via such a tin holder:
If power wires are long, it makes sense to take them thick - to make resistance (and voltage drop) smaller.
I re-soldered original lithium-ion battery from the Xeccon S14T light (it was 2S), so there's no protection board anymore. This is dangerous - lithium! (some other cautions on CPF). So I protected it with a fuse. I made one myself by cutting a very thin stripe from a flattened piece of tin (whatever they use) from a 20A-fuse.