With the stock 3rd brake light, the light goes on when you unlock your car. That's what that third wire controls. Without it, the third brake light only works when you press the brake.
I used the Oracle lighting spare tire LED, someone elses video here:
My install pics here
Then I put the American Adventure Lab Spare Tire Platform:
https://americanadventurelab.com/product/mass-spare-tire-platform-stp/