Not all encryption is end-to-end. Only public-key-cryptographic schemes are capable of providing end-to-end encryption, and that includes TLS (despite this article stating the complete opposite).
Yes you are right that the type of key doesn't matter. Only how the key is exchanged/distributed is what matters.
I meant to write something different (and I will correct that with an edit after this), I meant to write "public key exchange schemes". However the miracle of 'public key exchange' is basically synonymous with 'public-key-cryptography', so I think using the phrasing "absolutely not true" is misleading...
The example of symmetric keys, which you mentioned, does indeed rely on public key exchange protocols (for example, DHKE is a public-key exchange system designed for sharing symmetric keys).