I think people tend to talk about "having a personality" rather than "being a personality" (unless they're talking about celebrities), so from a language point of view I'd say it's the first.
However, if you dig past common usage then I guess you're asking whether personality is an emergent trait of the mind, or whether it's integral.
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What is "zapping"?
An irrelevant detail.
Also, can't you just ask...
Probably, but that's not how the question originally came up.
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However, if you dig past common usage then I guess you're asking whether personality is an emergent trait of the mind, or whether it's integral.