The properties missing from the default properties are partly due to the way MODX handles default properties (though they are sometimes oversights of the programmer).
Say that you have a property called 'published.' If the user sends it as a property, you want to use the value the user sent (0 or 1). If the user didn't send the property, you want to use the publish_default System Setting.
The problem is that if you put the property into the default properties, isset() will always find it in the $scriptProperties array, even if the user has intentionally left it out of the tag so it will be set via the System Setting.
There are also cases where you don't want to expose an advanced property to naive users. As a result, you leave it out of the default properties, but put it in the docs.
There are various ways of handling these issues, but all of them involve compromises.
All that said, there's no excuse for having the properties in the docs misspelled, but it's easier to do than you might think (since your own variable names often won't match the properties), and once you've written more than a dozen add-ons, it's inevitable.
I'm going to guess that isLoggedIn was a late addition that was tossed into the package as a result of a question in the forums -- probably untested. I think a lot of people use Personalize instead, since it's simpler and they don't care about contexts. I suspect that many users aren't even aware that the Login package contains isLoggedIn.