I believe the first three; identity, core self, and authentic self, are defined by us and our personality is what others perceive, from our behavior.

I identify with what I identify with! One hears/reads about men retiring and then dying because they have identified with their working selves and not developed any other interests and now that identity is gone. Similarly is women who primarily identify as "mothers" and the children grow up and they are at a loss how to behave, what to do with themselves.

If I were to look at one of these two people; I would perhaps see the guy had a "Type A" personality, was intensely into work. The woman I might see as mothering (or smothering :-)

Those are the stereotypical examples. Most people are more complicated than that. Your mother, for example, might also like to write or paint and identify with being a writer or painter/artist. The guy might belong to civic or professional groups and might identify with being a civic leader or Accounting professional, lawyer, or doctor. Some of those identities have their own stereotypical personality traits; the lawyer might be someone who loves to argue, we might say they have an argumentative personality. We might find an accountant to have a "bland" or boring personality.

However, my looking at the accountant and what he does, his outward self that I can see is why I might say he has a bland personality. He could though have a rich inner life that I never see or that he never discusses; he could be an arm chair traveler and identify with explorers (or pirates, football players, etc.). Maybe he was on his high school football team 40 years ago and really loved that period of his life and still identifies with it. I may still decide he has a boring personality. Look at Bob Newhart; he was an accountant who became a comedian; his personality couldn't fit itself into an accounting identity.

The "couldn't fit" part is where the "core" or "authentic" self part comes in. Bob Newhart is the only one who can tell where he does or does not fit! I can look at his personality, his always making jokes, etc. and suggest he become a comedian but only he can tell if that would be a good identity for himself.

I love working accounting problems but do not wish to become an accountant. I am here daily, love psychotherapy and the process of learning about myself and others but my core self does not allow me to be a psychotherapist. I would be miserable sitting for hour after hour listening to others, my preference is to listen to myself.

I am a writer and a researcher. I have written a novel http://mysharingspaceonline.com/story.pdf and spent the last two years doing intense genealogical and historic research. In November I wanted to write a novel about my 3rd great aunt (b. 1755, d. 1813) but had difficulty. Looking inside at my "core" self, I could not sacrifice the research, the "truth" of what I'd found and fill-in-the-blanks with fiction. I turned my novel writing into a research journal instead. It is that looking inside and knowing who I "am" that let me know which way to jump, that helped me solve my discomfort over trying to fictionalize my previous two-years' research. I could not do that. Technically I could; anyone looking at me and my self-identities and personality would say I could, but my core self said I could not and came up with what was best for me.

I pretty much use core self and authentic self to mean the same thing. I was being authentic in my choice of research over novel writing; I guess, technically, my core holds both the writing and the researching selves and it is the authentic self that chooses in any given need/situation?

Confusion happens if we do not know ourselves, what we like, what we feel, or if we let others define us. "Wanting" to be something, identifying with writers, say, and not writing, that either has to hurt our core self (why writer's block is painful to writers) or we are only fooling ourselves (not knowing our authentic self) and wasting time saying we identify with something we don't really. That is why it is important to look inside at what one really wants (identity) and feels (core) and then to express those things (authentic/personality).