Question: I'm trying to learn details on how LiveID works, when compared to other federation technologies.

To be honest, I'm a bit overwhelmed by all the options at <https://msm.live.com/> and want to understand what I'm doing before I federate my application to LiveID. 

In addition I'd like to understand the differences between the free implementatations and the RPS system. My understanding is that there are at least 4 different ways to hook up to LiveID. Considering that large surface area of authentication, and little public documentation, makes me uneasy.

Answer: 
LiveID is an extension of WS-Trust/Federation/Security in a passive mode. Passive meaning that the client -- the browser -- needs to be told what to do by the clients, which is done through 302 redirects and POST-backs.

To rephrase this, LiveID follows a Claims-Based model, building on top of the protocols mentioned earlier. The documentation is very sparse because, well, Microsoft can be very lame at times.

The differences between the versions have to do with how the authentication happens, moreso what layer makes the redirect to login.live.com. E.g. the server-side code forcing the redirect, or the client code -- javascript -- does the redirect.

What exactly are you trying to accomplish? Are you just wanting a person to log into your site using their LiveID, or are you looking for a more in depth federation model?
I'm not sure about this, but I believe that LiveID is not it's own set of technologies, but simply Microsoft's offering in public federation.  

They wrap it up nicely and provide easy interfaces, but underneath its still the same techs and protocols: ADFS, WS-Trust, SAML, etc. They might also give you some flexibility in mix-n-matching, that would be difficult to do on your own.  

I'm also pretty sure that LiveID is also OpenID-compliant, being both a provider and consumer.