Thanks so much for providing this forum!
Trying to bypass a controller on a 4.5 v Xmas tree array. My problem did not appear on your list of examples. Hope you can help
Making an assumption or two; my best guess based on the information and photos is the controller is a 'bridge' type. That is, the supply's plus and minus can be flip-flopped between the 2 black wires. The black wires do connect to the lights, yes? Wiring of the string itself would determine what colors or patterns are created. A bit more information would help such as a rear photo of the board and light patterns when the string was functional.