That's weird that the black is +.
The Germans (Porsche/VW) did that before though. Where brown is negative. Brown is earth, therefore it was ground.
Your link shows it has red, black and a white for the harness. The link also shows a grey (presuming black), red and a white.
With the switch and relay hooked up, do you get a black (positive) and brown (negative) output? Check with a meter.
The LED lights will only work with positive on positive and negative on negative. They won't work like a light bulb would if you switched polarity.
If you don't know which one is which on the bulb. Check with with a fused line directly to the battery. If it doesn't work one way, swap positive and negative to the LED. Then you will know which is which on the bulb.
I hope that makes sense.