Fixed it; was the starter relay.
I jumpered the relay as shown here:
The starter engaged fine. There was a radiator fan relay that is the exact same there near the starter relay, so I popped it out and replaced the starter relay with it. Car started fine, so I went and got a new one, put the radiator fan relay back where it belongs, and put the new one in the starter relay spot. Car starts up, and everything seems fine.
Here is the fuse/relay box under the hood looking at if from the driver's side fender:
All of the relays circled in green are the exact same model. You can see where I have already replaced the Starter Relay, and I marked the relay that I used to test it.