to answer your original question. they gave you the wrong part at the parts store, the thing in the picture is the only starter "solenoid", its actually a heavy duty relay but its called solenoid for simplicity, a starter solenoid goes on the starter motor, which these fords do not use.
when you are having a starting problem, does it go CLICK CLICK CLICK? or does it do nothing? you said the battery and alternator tested ok. that leaves your starter relay, the wiring, the starter motor, and the negative battery cable.