I'm a little fuzzy on some details, but my first experience was in about '73 when I lost my keys (or someone stole them) while I was surfing on the North Shore, at Pipe. My '64 bug had only three wires to the ignition switch. I vaguely knew that one was power (obvious, the big red one, exciting sparks when grounded), one went to the ignition, and the other to the starter. Now the curious thing to me at the time, reading my Clymer manual there in the Ehukai Beach Park parking lot, was that the wire to the starter didn't actually go to the starter, but to the solenoid. It activated the solenoid causing it to lever the pinion into the ring gear, which grounded the starter and cranked the engine. I was mildly surprised, but hey, only three wires, let's do it... short story, started right up and kept running as long as big red and ignition stayed together. Got home and used that technique for some time thereafter. The scenario later repeated years later when the keys to my '67 Squareback were stolen at Tonggs... I eventually fitted a toggle switch from an abandoned Bus, and a pushbutton starter switch bought elsewhere. One needs, one must. No one successfully stole either car, but I caught later one St. Louis High School punk trying to steal the Bug, and three guys together didn't get the Squareback push started though they had the ignition on. But I digress...
Recently my seldom-driven CS will loudly *click" when I turn the key, but not spin the starter. Several tries later, it engages and cranks the engine just fine, and it will start, as long as I've primed the carbs with a ounce of two of gas beforehand. If not, 20 seconds of cranking (I think too long, may overheat the starter) ensues before enough gas is sucked from the tank, fills the float bowls, through the accelerator pump while I'm stomping the accelerator pedal all the while. Once started, it will reliably start every day, as long as it's used every day. I read that this is not unusual for Zenith carbed cars.
I mention this for a little background on the starting sequence. The solenoid not only throws a lever to push the pinion gear into the ring gear, but grounds the starter (which we all know has a big line from B+, always hot). Once the solenoid causes the pinion and ring to merge, and if it grounds the starter, the starter will spin, crank the engine, and off you go. Assuming other bits are in good order.
On my car, the first few clicks of the solenoid don't cause the starter to spin. I think the starter isn't being grounded, though the pinion and ring gears may be engaging. I took out the starter once, a real humbug job b/c I had to remove the rear intake manifold and carb. The starter isn't sealed, but you have to de-solder a connection at/on the solenoid. I didn't want to do that, so just bench activated the solenoid and starter a bunch of times, separately and together, sprinkled holy water (and some WD-40) in dark places until things seemed to work. Reassembled, etc.
Hypothesis 1 is that the solenoid isn't throwing the pinion gear rearward (or far enough) to engage the ring gear. This suggests a worn area on the ring gear where the pinion makes no contact at all, as shown in Toshi's post 11 above. It may still start if in good tune and the pinion catches where there are still teeth. Hypothesis 2 is that the starter isn't reliably grounding.
My battery is years old and I'm surprised it still has juice. Never recharged, seldom used... but once it successfully starts, no problem until months later when it has gone sleepy again and the sequence recurs. Make of it what you will, but for the humbug of getting the original starter out, perhaps $70 for a stronger replacement isn't a bad idea. Sadly, shipping across the pond (starters can't swim) means that I'd pay a good deal more...
