Symmetric cryptography is like a door lock. Everyone that has a key can lock and unlock.
Asymmetric cryptography is like a common lock or handcuff. Anyone can lock it but only the (private) key unlocks it.
I assume that "private" here doesn't mean "just for one person", rather that it means only-unlocking key but then, instead "private" wouldn't just using broader terminology be more accurate and elegant:
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Is the term "only-unlocking key" more accurate than the term "private key"?
Asymmetric cryptography is not about locking or unlocking something. While the metaphor might make some sense for encryption (public key) and decryption (private key), it does not make any sense for signing (private key) and signature verification (public key).
Apart from that "only-unlocking key" and "private key" describe different viewpoints - one is about what is done with the key and the other how the key needs to be protected. Both are valid viewpoints and none is "more accurate" than the other.
Private key is named private to stress that it has some properties only as long as it is kept secretly.
What properties it allows to maintain?
Confidentiality: As long as you keep it secretly, only you can decrypt messages encrypted with the public key.
Authenticity: As long as you keep it secretly, only you can use it for signature, nobody else can sign anything in your name.
