Another name for an apocraphal story could be a unauthenticated story. Apocraphal stories are often not part of the official or recognized canon. They might be legends or rumors that have been passed around without solid evidence to support their truthfulness. So, unauthenticated story is a fitting alternative name.
A apocraphal story can be named as an unsubstantiated story. Such stories don't have enough proof or evidence to back them up, just like when we say a claim is unsubstantiated. They float around without the necessary factual basis, much like rumors that may or may not have any truth in them.
Well, 'the commencement' is another way to refer to the start of a story. It's like the story is about to commence, start its journey, and draw the reader in from that very first moment.
The start of a story can also be called the 'opening'. For example, in many novels, the opening sets the scene, introduces the main characters or gives a hint about the main conflict.
A yarn can also be another name for a story. This term is often used in a more informal or folksy way. People might say 'spin a yarn' which means to tell a story, usually one that is interesting or perhaps a little exaggerated.