I am aware that the earliest fluorescent lamps were all designed for preheat use (obviously). When rapid start was developed, there were special rapid start lamps made that were somehow different from the preheat lamps. Not long after that, the modern "preheat-rapid start" lamps came out and are made to this day (though they are no longer labeled "preheat-rapid start"). My question is this: what's the difference between the three types? Cathode heating voltage? What happens if a preheat-only lamp is run on rapid start and vice-versa?

I really don't know of any difference; I've had lamps marked "Rapid Start" running on preheat or manual preheat (22w circlines in prehear adapters and and a 90s Sylvania Workshop F40 on a manual preheat then once in an "automatic" preheat fixture and they were fine.  I believe the cathode voltage for 4ft  T-12 lamps is around 3 volts, but for stuff like F20T12s it's 9 volts, though again I see no difference between preheat or rapid/trigger start voltage for either on their respective preheat or rapid start ballasts.

For some RS lamps become necessary to use an internal conductive coating in order to keep the ignition voltage low enough, mainly makong it less sensitive to outer surface wettness.

Preheat starting does not need that feature, as the generated ignition spike is high enough to ignite the lamp in virtually any state (cold, humid, dirty,...)

I don't know about the PH 40w vs. the RS 40w, but on the smaller PH lamps, along with the old F90T17's, the cathodes won't heat up enough with the ~3.8v on most RS ballasts. That's why there are "trigger start" ballasts....being that the filament voltage is closer to 9v for proper cathode heating. On some of the earliest ones (like early circline or 20w) the filament voltage was as high as 15v! If lamp didn't strike, it would look like a stuck starter and would blacken ends rather quickly. At some point early on, they reduced that to 8-9v and called it "trigger start" for what ever reason.

The correct heating voltage for the RS isn't just the voltage able to heat up the filaments, but at the same time not overheating it during normal operation. And there the 3.5..4V is really the maximum.
What I've measured (on the European market lamps, so preheat) was, at 3.5V their resistance reach exactly the 4x the cold resistance, a value very frequently cited as a sign of a correctly heated filaments for an ignition.

When the ballast design does alter the heating voltage so it does not exceed the 3.5..4V when running, there is virtually no upper limit for the voltage used for the initial preheating phase, provided it is reduced back below the 4V before overheating the cathodes (CFL ballasts use about 20..30V, therefore the heating time is just fraction of a second).
One of the easiest method for this is to tie the preheating to the presence of the arc: Once the filaments reach thermionic emission, the arc is ignited and so it's presence stop the cathode heating power (you may design the RS transformer to act like that, feed the primary of the heating transformer from the voltage across the lamp, resonant current in simple electronic ballasts,...). I would guess the "trigger start" is exactly a ballast of this type (in the "magnetic" version)

Other method is then to use a timer for that (all the modern "programmed start" ballasts).

When I worked in a factory they used the same type of tubes in both the preheat and quick start fittings it didn't seem to make much difference