1984 EZGO Wiring

Evening all! I'm new to this site and just got my first cart. I sort of hijacked another thread for my first post but I'm trying to make that right now. I'm a veteran at small engines both 2 and 4 cycle. I haven't however had a small engine that stops/starts based on pressing the pedal and some electrical things. So, I'm trying to sort out the wiring on this cart because it was apparently truly messed up by some teenage boys who didn't know any better and a dad that refused to fix it following the disaster. I was told the solenoid failed and the boys tried to replace it. This led to me being able to trade a broken 93 Dixon mower for this cart.

So to the point. This is a diagram of the wiring as I have found it.
(Please see my photo album due to me being a newbie and unable to post pictures in my post)

This is a picture of the solenoid that was put on the cart.
(Please see my photo album)

Two questions here. First, does this appear to be a proper solenoid for this cart? Second, if so how do I relate the S and I posts to the wiring diagram I found in the manual for this cart on this website?

I haven't put a battery on it yet to start testing things. What size/type battery should I get? I didn't find it listed in the manual.

I'm also assuming that I need to redo the wiring to match the diagram found in the manual?

The manual I keep referring to is 19980-G1

Thanks for all the help in advance! I look forward to joining the golf cart group. Hopefully soon I'll be able to contribute to helping others around here.