I think there is a misunderstanding that because they are each a "first step into cave diving" that Intro, Apprentice, and Cave 1 are the same program. They're not.
Before I try to illustrate the differences, let's start with a quick discussion about the final outcomes of all cave programs -- Intro, Apprentice, and Cave 1 were never meant to be end points -- the final end is FULL CAVE, period. This comes back to the very first rule of accident analysis -- untrained and incompletely trained cave divers have a poor safety record in cave environments. No amount of openwater experience can properly prepare anyone for the cave environment, heck, even OW instructors have a poor track record in caves. I can give several examples of OW instructors that died cave diving because they didn't know what they didn't know.
So let's accept that Intro, Apprentice, and Cave 1 were designed to each be a stopping point to refine skills and let everything sink in before proceeding to the next block. Let's also accept that if a person is going to regularly cave dive, they need to complete their education and work towards being a full cave diver eventually.
Now, with an understanding that "Full Cave" is the eventual outcome, let's look at the journey on how we get there. As I've said before, Intro, Apprentice, and Cave 1 are different points on that journey and are representative of the amount of training that has gone into the total trip. The differences in the amount of training that has gone into each stop is why Intro, Apprentice, and Cave 1 have different limits (1/6ths vs 50 cubic feet vs 1/3rds).
To illustrate where each sits on the journey towards Full Cave, I'm going to use a metaphor of a 10 mile hike with 0 being OW Instructor (or AOW diver or whatever, zero overhead training) and a full cave diver with stage being mile 10.
Mile 0 - AOW Diver, Divemaster, or OW Instructor - someone with zero training for the overhead.
Mile 2 - Cavern, Intro to Tech, or Fundies. Usually 2 to 4 days, outcome: Understanding of tech/cave gear, propulsion techniques, buoyancy, trim.
Mile 3 - Intro to Cave or old NSS-CDS Basic Cave. 2 days past Cavern/Intro to Tech/Fundies. Outcome: Tying into the gold line, most basic penetrations (1/6ths).
Mile 4 - New NSS-CDS Apprentice. 4 days past Cavern/Intro to Tech/Fundies. Outcome: Tying into the gold line, most basic penetrations but a little more liberal on gas (up to 50 cubic feet, or 1/3rds, whichever is less).
Mile 5 - GUE Cave 1. 5 days past Fundies. Outcome: Extended penetrations (1/3rd of 2/3rds) and navigation beyond one permanent T (and multiple gaps but no jumps).
Mile 6 - NAUI Cave 1 and old NSS-CDS Apprentice. 6 days past Intro to Tech/Fundies. Outcome: Extended penetration (1/3rds) and two navigational decisions (jumps or t's).
Mile 8 - Full Cave Diver. 8 days past Intro to Tech/Fundies. Outcome: Extended penetration (1/3rds), unlimited navigational (jumps/t's/gaps), circuits and traverses.
Mile 10 - Full Cave + Stage, Cave 2 (both NAUI/GUE). 10 (or more) days past Intro to Tech/Fundies. Outcome: Extended penetration (1/3rds) with stage bottle use.
As you can see, the amount of time to get to that first step is different between the agencies and that's why the outcomes (what the diver can do) is also different. There are various philosophical reasons why those first steps are different, but at the end of the day, they all are in basic agreement that the final step is supposed to be full cave and that the first step is simply a stopping point on the journey.
Whew, this is longer winded than I had expected... Hope it makes sense.