Classical myths from antiquity are literary versions of what was apparently oral narratives from contemporary folklore. Many of them were apparently legends - stories generally told to be believed. Some appear to be historically related to folktales gathered in the nineteenth century, but it is unclear if the stories functioned like pre-modern folktales - stories told as fiction, much like oral novels. See below for definitions.

It is too easy to refer to "other people's religion" as mythology, so one needs to be wary of the term and its negative connotations. The thing is - when attempting to answer your question - is that all cultures have legends of some sort - stories told to be believed, as I indicate. So there is no absolute point where a culture's legends are recognized as fictional/fantastical/unverifiable. One must ask ... which culture? And, what myth? It is easy to look at other people's legends or legends of the past to recognize them as unbelievable, while at the same time embracing one's own legends and folk beliefs.

I suppose you're asking about Greece or Rome - that's usually what one means, but one must remember that there could be a world of difference between a philosopher considering the oral narratives told and embraced by an illiterate farmer in the field, so finding the "point in history where 'myths' came to mean" false is a moving target that is very different depending on the observer.

For definitions see the following comment...