I realize the original posted question is quite old, but with a few recent replies, I thought I would chime in as I do have first hand knowledge in this area. I graduated BUD/S Class 209 in Feb. 1997. A few points:
Second phase of BUD/S (aka "dive phase") taught us the basics of diving open circuit as well as closed circuit. Our first week (of nine) was very basic stuff with open circuit twin-80 rigs. The following 8 weeks (for the most part) focused on closed circuit diving with the Drager Lar V. While there were still evolutions with the aluminum twin 80's as well. We DID have to learn the ins and out of not only what it is good for (Lar V and open circuit), but also why and HOW it works. Couple all of this in with dive physics as well as physical training and it makes for a very tough 9 weeks.
It is the most academically challenging phase of BUD/S with the dive physics but also a lot of fun. While it is only 9 weeks long, it is 9 solid weeks of nothing but diving. Dive, dive, dive. If we were not in the classroom, we were at the pool, the bay or the ocean.
BUD/S is the first step. By the time we get to 2nd phase, a student has proven to be a very capable combat SWIMMER. 2nd phase tests if he has the ability to be a capable combat DIVER. It does not teach a student everything there is to know and the focus is always on the use of the different rigs from a military perspective rather than a recreational one.
To address a few other points that have been brought up. As noted, the Draeger Lar V is NOT semi-closed. It is a closed-circuit RB which uses aviation (99.5% pure) O2. We did get to work with limited semi-closed circuit exposure with the Mk VI rigs. The benefit was deeper depths than capable with the fully closed Lar V. However, it does still produce some bubbles. It was more for familiarity as it is rarely (if ever) used in any practical sense anymore (I have not seen it used beyond that). It's either all open or all closed. But semi is out there just old stuff, usually based for SDV work (at least from its inception that was the plan).
As for people claiming to be SEALs. God, I could fill about 5 notebooks full of the bs stories I have heard from people claiming our status. It is usually easy to weed them out with a few questions though. I don't know what it is the brings out the phonies (Hollywood I presume), but they are all over and that is a big pet peeve of mine, but that's another topic.
Back on point of the original issue - as I stated BUD/S is step one for a SEAL. There are 6+ more months AFTER BUD/S that must be completed BEFORE the Trident is earned. I graduated from BUD/S Feb 1997 and did not get my trident until October of 1997! There is a lot more to learn about diving beyond 2nd phase of BUD/S, but even in those 9 weeks, there is a lot of information and we were very capable divers even at that point. That was my very first exposure to diving. The approach taken is complex but the weak are weeded out quickly leaving those that are dedicated to learning and developing our craft.
We do not use diving for simply inserting and extracting operationally. We have operations that rely solely on underwater work (not many in the recent world situation) but those operations do exist.
I could on for pages with this thread but I'll end it here. If anyone has any specific questions regarding SEALs or BUD/S feel free to ask. I love talking shop. Thanks for letting me chime in.