If you jump to the smaller post on the Mean Green starter now, the starter should start up. It probably will, but this is just a sanity test to verify that you brought home a good unit from the parts store. Before you go much further, you should take note of the fact that the Mean Green starter lacks a nose cone bushing, which makes it an inherently weaker design. There is a reason that many people stick with OEM design components. Engineering is design, testing, testing, testing and more testing. From economic and scale considerations, there is just no way the Mean Green design has received the same amount and quality of testing as the OEM design.
From what you have posted, the brown wire is continually energized, whether the key is on or not. It should be easy to verify this with a test light or voltmeter, another sanity test. This could be caused by the key switch, the relay or a short in the wire. The relay sounds like the easiest thing to fix. Many times we go with the easiest fix first, but you have to decide whether you can throw away the price of a relay in case you are wrong.
Earle