Ed,
As saved is not to my knowledge a setting available from within the PSE, it is configurable via load options. That would appear to be one main difference. Precise basically refers to the ability of teamcenter via the PSE to lock an assembly so that it loads a specific revision of the components. The two most frequently use settings are "Working" or "Latest Released".
If you're a new user to Teamcenter then expect to be confused about the control you used to have via "Load Options" being in part transferred to the PSE, and about how they interrelate.
I couldn't tell you much about Obsolete or Sharable except to say that they sound like inventions of your own customization rather than out of the box settings that I'm familiar with from elsewhere. The idea of whether a file is sharable may in concept be achieved without creating a specific status to manage who sees the file. Sites I have seen create accounts with working groups so that some groups files are visible only within the group and baseline data is either published in a restricted or general sense. I'd be digging a hole for myself to say that I knew how it was really done, but I have worked under those systems and it appears to be how the product is intended to be used. I'd simply advise you to look to use the product in the way it is intended to be used and avoid over customization, so that from one release to the next you may have a less traumatic time in the future. Any thing you do early on needs to be well thought out because you immediately start to create data with a legacy that may need to be maintained or reworked to fit with the system you eventually settle on.
On your final point I think you may be correct speaking only from observation. As with any system some user input is expected to drive the system to work properly. Sometimes the inputs required don't fit entirely with our expectations and I would in that case query with your local help desk whether there is a setting the you aren't getting right. That is to preface the answer that I believe it works properly from the PSE but I have noted that when loaded directly into NX that new versions aren't always loaded as you might expect. My suspicion is that this may be defined as "a feature not a bug", since it provides that the owner of the assembly controls how it is maintained via the PSE, rather than other users doing so on an ad-hoc basis.
Personally what I see is that most design work is done in the pre-release stage of the product life. Many of the features that do well to maintain systems under change control after drawings have been released seem to get in the way earlier in the process. Teamcenter as I see it tries to strike a balance. It is a very powerful tool with a lot of sophisticated features all of which most of us don't frequently use or understand particularly well. Occasionally controlling that almightiness requires more work of us to maintain than we might welcome.
Simply put I would invite a simpler and quicker way to baseline components and their assemblies with fewer steps and less waiting for the operator, by way of having the system take on more management of these processes, preferably in the background.
Hope this helps
Hudson