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:
- locking and unlocking key
- locking-only key
- unlocking-only key