The pictures are a waste of time and a useless distraction. FOCUS!

You describe the two pairs of wires as though they were grouped together, This disagrees with available wiring diagrams. ARE THE WIRES ENCLOSED WITHIN A MODULAR CONNECTOR OF SOME SORT? Or are they separately and individually soldered, which they should not be?

Examine the bike carefully for a charging port. About as big around as your thumb, three prongs or holes, with a cover of some sort. Also look for a large rocker switch. One of these should be on the frame.

Does the bike have any sort of factory installed lighting? Or user installed with no visible power source?

Also, have you located the access panel for the controller? Should be on the downtube, down near the cranks. Controller is custom, internally mounted, replacements likely not available, as with many parts on this bike.