Earlier quoted context omitted.
You're probably confused by "SHA-512/256", which does not mean SHA-512 or 256, but rather a truncated version of SHA-512: https://en.wikipedia.org/wiki/SHA-2 in the third paragraph.
So why would a truncated version of SHA-512 be better than SHA-512? And why is SHA-512 = SHA-256?
Length-extension attacks are relevant when you design a MAC by passing a secret and then a message to a hash function, where only the message is known.
Truncating the hash (which is what SHA-512/256 and SHA-384 do to SHA-512) removes the ability to grab an existing hash H(k || m) (where k is unknown and m might be known) and append junk because a truncated hash does not contain sufficient information to recover the full state of the hash function in order to append new blocks.