But what is that "solenoid slug" and that "solenoid coil" doing in there? And why are they called that?
Because that's what they are.
But if it's got "solenoid" parts inside it, could that be precedent for calling this stupid thing a solenoid again?
No, it's actually unrelated to those parts.
A toaster has a several heating coils of resistor wire in it, and a thermostat that mechanically releases a spring. But it's just as wrong to call a toaster a "thermostat" or "heater" or "spring" as it is to call a relay a "solenoid". Same for calling a truck a "transmission" just because there's a transmission in almost every truck. A relay is a switch that's operated by a solenoid (so that the switch can be controlled by another electric circuit). But a relay isn't simply a switch; and it isn't simply a solenoid, for the same reason that a solenoid isn't simply an "electromagnet", even though every solenoid contains an electromagnet.
But the reason we're having this discussion is that most automakers (including Ford now) started off using starter motors that had integral switches operated by complicated linkages that allowed a foot pedal to push the drive (Bendix) out against the ring gear before that switch closed to spin the starter. Later, the linkage was replaced by the wires between a small dash switch & a solenoid on the starter, which first pushed out the Bendix (just like the older pedal), and then closed the switch (just like the older pedal) making it ALSO a relay, but still primarily a solenoid. Ford went a slightly-different direction, and kept the relay remote from starter, but used one of the motor windings as a solenoid coil (because they're both electromagnetic coils) to pull one of the starter's pole shoes (now acting as a solenoid slug) down to push the Bendix out via a fork (just like the solenoid starter's). When the shoe was all the way down, it closed another switch (integral to the motor windings) that energized all the other windings, making the motor spin. Here's the exploded view of that starter, which is easily-identifiable by the angled lump on the side (the cover for the raised pole shoe):
But because the true-solenoid starter was more-common, the (lazy, uneducated) parts monkeys got used to saying "starter solenoid", even when they were just selling the switch that went on the back of the solenoid. And they passed that bad habit on to everyone else. Here's the PMGR starter with the cylindrical solenoid on the side (to pull the fork) and the switch on the back (to turn on the motor, only shown in the cutaway at the top far Right):
So when a Ford-owner asked for the switch (relay) that turned on his starter motor, the parts monkey told him he was buying a "solenoid". So it's not the fact that a relay contains solenoid parts that caused the confusion - it's just lazy partsmen. The fact that Ford switched to PMGR starters WITH true solenoids in the 90s only adds to the confusion, but even those vehicles STILL use a starter relay in the same location as before (near the battery).
That's a whole other kettle of worms... But those are mainly used for 220VAC or 3-phase motors (or other inductive loads). That term doesn't really apply to 12VDC vehicles.
...the part manufacturer calls their part...
Look again... %) The part manufacturer is Chinese, so he doesn't call it anything English. He just prints the word on it that BWD (a parts company) orders him to print on the box.

But even Ford is guilty of that kind of misnomer, for much the same reason. It's easier for inventory management to continue using a name, even if the part design changes substantially. Look at the names on these boxes:
Coil Kit, Inflatable Restraint
Contact Assembly, Clock Spring
They both contain a coiled component, but neither one is a "coil". They both contact other parts, but neither assembly contains any electrical "contacts" (which are not the same as electrical "terminals"). And neither has anything to do with a clock. And I doubt either one contains a spring (none of the Ford ones do). But they're nearly-universally referred to as "clock springs". Ford usually calls them "airbag sliding contacts", but that's not what they are - it's what they REPLACED when the technology advanced. But the name stayed for many years after that advancement.
The point is: the name on the box is not the defining characteristic of a part. A part is what it is, and this thread is about a starter
RELAY.
