Only problem with the starter relay in that pic is it's not a starter relay, that the head light pig tail, It goes on the headlight bulb and hooks into the wiring harness.
On your starter RG has the most probable cause.
Once you get a good battery it should crank.
Next I would try Jumping across the to large terminals Where the battery cable and starter cable hook to your solenoid. This should crank the engine. If not it could be a weak ground path from the starter back to the battery. Or possibly a weak connection on the hot side.
Leo