Yep, the white thing unplugs from the motor, as you have discovered. The wires are obviously not meant to be able to come free of the plug though. Have C3D support not gotten you a new one? If so I'm very surprised!
I don't know if this will help, but I fixed mine by stripping the insulation off that yellow wire a bit, tinning the end with some solder, then touching the iron to the little metal collar inside the white plug and kind of jamming the yellow wire back in there. Took a few tries, and scorched up the plastic a bit, but once I got it in there it seemed fairly secure. I tested for continuity and then put a few drops of hot glue on to help hold things together. It does not look nice, but it works perfectly, especially with the strain relief.