Green electrical connectors
 

Tried a search, but....

On the D/S, behind the battery I have 2 connectors. One is a 4 pin, one is a 2 pin. I am looking at tapping into the 4 pin which has a socket that powers up in the "run" position only. Is this okay for running my driving lights to be on only when running?

The four pin one is the diagnostic connector, and the 2 pin is the initial set connector. Top center pin of the 4 pin connector is 12v with key on but personally I would find a different source for your lights.

It'll be fine to operate a relay, but definitely not the lights themselves. Power the lights directly from the battery via a fuse and the relay.