Well you're right about the confusing part. The red wire is 12v,because it's prior to the resistor. Then the black wire would be reduced voltage,using a ballast resistor or coil with internal resistance.
It's been 5 years since I hooked mine up and at time I also ordered their coil. I didn't make note of weather or not that coil had internal resistance, just hooked it up using no ballast resistor.
From what I see, it would be necessary to use a coil with resistance or a ballast resistor.
Neswes, to answer your question, Don't use the ballast resistor if the coil has internal resistance.
Now, does that flame thrower have internal resistance, don't know, but my guess would be yes, but you will need to determine that.
Tim, thanks for pointing that out.
Dan