The term relay is pretty well defined as an electrical switching device that's controlled by a solenoid. Strictly speaking, a solenoid is a winding of electrical wire in a cylinder, and is a type of coil. It generates a strong magnetic field aligned with its long axis.
https://en.wikipedia.org/wiki/Solenoid
Relays and contactors are pretty much the same thing, with relays being smaller and contactors being larger. Both contain a solenoid that switches the contacts, because the solenoid makes a strong local magnetic field that can close the contacts quickly and securely.
Solenoids are strong and can do different types of work. The SS581 contactor I linked above would be better called a contactor, rather than solenoid. It has a solenoid coil in it, but solenoids can be used for other purposes, like the valves in your washing machine that are solenoids (solenoid valves).
I think the use of "solenoid" for the starter contactor in an automobile is historical and somewhat anachronistic. Maybe relays were widely called solenoids when Kettering invented the electric starter ... dunno. Relay, contactor, "solenoid" ... in my mind, all different names for the same device. Up to the reader to interpret based on context.
Now, there's a lot of ambiguity here that must be resolved by the reader. You say that Prestolite starter has a solenoid in the starter body that bridges contacts and runs the starter motor (like the Ford type SS581) and throws the starter gear into the flywheel. That's a solenoid in the sense of being a contactor, but it's also a solenoid that does mechanical work to move the starter gear. The Ford starters have a similar scheme, but use the field coils (not solenoid coils, instead field coils) to pull a big steel slug on a hinge into the field, and throw the starter gear.
In the Ford starter, the field current both throws the gear and powers the motor after switching on the armature. Looking at your link, the Prestolite has to have both the solenoid and the motor field. I assume that there are contacts in the Prestolite that switch both the field and the armature on, once the gear engages with the flywheel.
Per your question, the solenoid must remain on while cranking. It works against the spring which would withdraw the gear if the coil were not energized. Solenoid off would also shut off the starter motor, I presume. The R terminal connects to the battery terminal when cranking, not the solenoid. When the solenoid is off (not cranking) the R terminal is disconnected from the battery. The solenoid does not connect to the R terminal, and the R terminal is open when not cranking. Thus it does not matter that the ignition energizes the R terminal when running, since it's open then and not connected to anything.
Not sure about the Prestolite starter, but I expect the "R" terminal is used to bypass the ballast resistor when starting. This gives more juice for starting, and improves ignition performance while cranking. You could do the same thing with a $5 relay controlled by the starter switch.
https://www.parts-express.com/12-VDC-Wa ... quantity=1
This rambles a little since I'm writing it quickly and off the cuff. Welcome to follow up if somethings not clear.