Well, the starter relay is called a "solenoid" in well over half the Ford manuals I have, also on the online parts sites. That's because it IS a solenoid, has a thick copper disk on the core inside that shorts out those big copper terminals to supply starting current. On later versions, there are various other smaller terminals running to other bits and bobs. At one time these solenoids/relays could be taken apart, the contacts reversed in their holes in the case and the disk flipped to get more life out of them.. Riveted together now, good luck with it! If your time is worth anything, get a new one. They run $10-$20, depending on who's doing the raping and where the things were made, normally not the US now. Can be had off the inner fenders or firewalls of most Ford vehicles in the wrecking yards for cheap, they haven't changed the design much in decades, used one of two configurations on most vehicles. The engagement solenoid on those starters so equipped is also called the "starter solenoid", so if you're looking for parts, you have to make it clear which part you're looking for. Ford did make starters without an engagement solenoid, they used a pivoted pole piece that flipped the starter gear out via a bellcrank arrangement. Has a hump on the side, the pole piece cover, instead of a tubular solenoid casing. Haven't seen one of those in years, though.
The O.P.s primary problem is the starter relay/solenoid, but if it's going, the starter might not be far behind. Brushes do wear, bearings do give out. Had one quit in the grocery store parking lot, got out and gave it a hammer rap, that jarred the brushes enough to start it one more time, next stop was the car parts place, left the engine running. Got to listen to what the buggy is telling you...
Stan