Look 1/2 way down this page. I think the manual is in there...
http://www.kymcoforum.com/index.php?topic=495.15Kymco works weird on the wiring... I THINK it is the rear brake switch activates the power to the rear light. That power is the one that is reading ground because it is reading thru the filament to ground. When you pull the brake lever that will/should light up hot.
The start button is the ground. You have to pull the brake, and have the tail light lit before the starter button can ground the relay.
If the relay is bad just get a Bosch or Cole Hersey 30 amp relay, and make new connectors for it. That will save you about 40 bucks over the Kymco relay that fails in a few years.