you have a lot of questions so let me answer this step by step:
You are right there is a difference between the closing the whole project
( as described on page 39 in the PMBOK " Characteristics of the Project Lifecycle")
and the project phase ( discribed on page 41 in the PMBOK " Project Phase")
To give you an real world example in terms of closure: I have worked on an aircraft modification project where we had different phases in our project. We were just in the middle of the "Design Phase" as the customer stopped abruptly the negotiated cyclic payments .
So after long time dealing with the customer back and forth the decision was made to put the project "on hold".
So to stagger the payments was already a lessons learned from one of the previous projects as the customer was bankrupt at the end of the whole project.
So comming back to my example the "Design Phase" was closed and we gathered the learning so far. After some month we had luckely a "ramp up phase" due to new payments and continued the project. Otherwise we had had to close the project and all procurements...
The process 4.6 "Close Project or Phase" is discribed on page 100 onwards in the PMBOK and when it comes to
Historical Information or the Lessons Learned keep in mind that you want to "save" the new knowledge which you have experienced in the "real world project" so far to improve all upcoming projects. You want to transfere it to the lessons learned knowledge base for use by future projects or phases. This can include information on issues and risks as well as techniques that worked well that can be applied to future projects.
At the time of project closure, there may be certain reasons for closing such as Phase closure, Project Closure, or Project closure due to early termination due to any cause (Force Majeure, Bankrupt etc...). Though all these happen in the Close Project or Phase, I understand there has to be a Administrative Closure for the project's early termination? What could be some common scenarios to usually consider FIRST after a project's early termination? Also any pointers on what goes to the Historical Information or the Lessons Learned? Or whether they are subsets? Am I missing something?