I would use a long traily lead as your reference, then you can determine if each wire is ok (do check for shorts to other wires).
Without seeing the system in the flesh it's awfully difficult to diagnose, ignore wire colours other than for finding the same wire at the other end, they mean nothing as to function.
Your unknown white, could be a neutral feed to "something".
First check the continuity between the light fitting and the known "other end" of each wire. Start at the light that works and work along the circuit until you find the break(s).