Is there a difference between "X itself" and "X proper"?

Question

Today I learned a new usage of "proper" in the form of "X proper" which means, according to Cambridge Online Dictionary, "belonging to the main, most important, or typical part". For example:

It's a suburb of Los Angeles really - I wouldn't call it Los Angeles proper.

I came across this usage when I was reading some technical document which says:

(Page 27) Compilation can involve up to four stages: preprocessing, compilation proper, assembly and linking, always in that order.

In fact, before I turned to the dictionary, I read another post that asked a similar question and someone replied:

Something you can do in English is, when referring to something in particular, you can say "the X proper" or "the X itself." Sorry, I speak English as a native language so my ability to explain nuances like this are about nil.

My questions are:

1. Is there a difference between "X itself" and "X proper"?
2. If they have no essential difference, which is more frequently used? "X itself" or "X proper"?
3. If they have some difference, which one should I use under what circumstances? I've learned English many years but I have never come across this use of "proper" until I read the technical document today, so, for example, is "X proper" mainly used in a technical/academic environment?

Answer 1

A search term is 'proper postpositive'. This has been discussed in other answers:

proper -- can this postpositive adjective be substituted for "in and of itself"?

https://english.stackexchange.com/questions/354077/use-of-proper-as-a-postpositive-adjective

From wiktionary:

(usually postpositive) In the strict sense; within the strict definition or core (of a specified place, taxonomic order, idea, etc).

Which matches the Cambridge definition you mentioned:

"belonging to the main, most important, or typical part"

It's the "core" part of something. The object in the strictest sense of the word.

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The word "itself" is more general, and can be applied in multiple ways, but the main sense here is "The object, the thing, 'by itself', separately, and not dependent on other things."

"Itself" is closely related to "himself", which could be used in an example:

He finished the repairs (by) himself.

That is, without assistance.

If A and B are clearly different things (already), and we say A itself (without B) , that means A separately from B, and without dependence on B.

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Going back to "proper", that's referring to the "core" essence of something. The real city, and not the suburbs.

There is similarity between "proper" and "itself". The word "itself" is used to point out a separation and a difference between two things. X and not Y. But usually in the case where X and Y aren't being confused with each other. "Proper" is used to distinguish between the "core main" part, and the "extraneous extra" parts, and when perhaps those two things were confused with each other.