I’m often mystified by such injunctions. Since I AM myself, isn’t “being myself” kinda inescapable? I’m with some of the Buddhists, who see the ego as a social convention and any “core” as illusory. I’ll leave the cores to the apples.

i have lived a very varied life...am i that abused child? the young battered wife? a mother? that person who loves dogs, gardening, travel? Survivor of numerous medical disasters & procedures? it ALL depends on where i was/am and what i have/had to cope with.
The young woman who escaped severe physical & emotional battering Before shelters & hotlines is not the same person i am now, but yet she is, and an abiding source of pride & strength. i would Never want to erase or forget her...and i do not think that is narcissism in any way.
However, she does color all my interactions & outlook(s) today in a way that may or may not be entirely functional 50+ years later, but...too bad so sad for y'all around me. (Being myself!)
I am, like everybody else, the product of my experiences & environment. To classify someone who acts/reacts based on their history & experiences as a ''narcissist" when things don't go your way, because of things you have no knowledge of in fact, is IMO just hurtful and pretty counterproductive, too.

The injunction to be yourself is often uttered by those who perceive someone to be identifying with an imaginary construct that is not in accord with or germane to the present circumstances or situation. Putting on an act is another expression that refers to someone is not being himself or herself. There are many ways to look at it, however, being yourself does not absolve anyone from responsibility towards others, it is not an invitation to act out whatever thoughts pop up in the imagination.

As with so many things, I think that it is the path through the centre that is the best way. Take either extreme and you will cause harm.

I think both the self (core of the individual as you put it) and a "web of relationships" both exist (individualism and collectivism there so to speak). Given the choice between those two, I'll take the former any day as the latter all too often has a way of eliminating any sort of personal identity. Unchecked self-actualization (narcissism and egotism) can lead to no good, true, but there's nothing morally objectionable to be yourself and forge your own unique identity.

That said, it's been my personal observation and experience over the years that the authoritarian control freaks in the world disdain those who desire to be true to themselves and their unique identities, and often times said control freaks will accuse their perceived opposition of being bigoted and or dealing in absolutes, as a way to avoid their own narrow-minded assertions. The group think mentality holds more destructive potential than does the individual mindset (think how cults are formed for example). That's my take on it anyways, and I could always be mistaken about such, but direct personal experiences and observations have led me to such a conclusion.

I think it means not to be false and/or a liar by agreeing with everyone and everything. Copying another's style means they haven't asked themselves what they actually believe and most coast by without any real thoughts of their own. I think I'd blame that on lack of reading. Reading teaches so many things one can't learn anywhere else.. If they can't successfully follow an idea to a logical conclusion something is wrong somewhere and should be fixed.