Long story short, I blew the transmission in my s2k 8 days before I shipped off to army basic training, after returning home and putting in a new transmission, the car had been sitting for a while so the battery went dead. Anxious to get her started up, I grabbed the battery out of my integra which I now know to have the dreaded 51R battery type where as the S2000 has a 51 type. The only difference between these batteries is the polarity but me being me, I failed to check the polarity when installing the battery. I recognized what was happening shortly after because the dash would not light up after I turned the key a few clicks, and switched the polarity on the battery. Due to corrosion on the battery connectors, the dash would not light up that well, and after further cleaning the dash lights up just fine, but when I try to start the car, a few lights on the dash dim significantly (but not all) and there is a click, and the secondary air injection system turns on, after releasing the start button, the secondary air stays on and the lights come back on in the dash. I checked the fuses relating to ignition such as fuse 3 under the dash, and fuses 41 and 42 under the hood. All three were good. My theory is that the poor connection due to corrosion on the battery terminals caused the 100 amp fuse (41) not to blow. What should my next troubleshooting step be?

Any and all help is appreciated.
Thank you!

Does the starter motor/ solenoid have diodes that are vulnerable? I’d really like to know before I go messing with it. The starter is such a pita to get to in this car.