I would say this is really complex question, maybe more philosophical than historical. Question about precise myth that was debunked at some point in history would probably give you more historical information about why this one myth was debunked. With this question you are asking when myth started to separate from history, and I have more questions than answers: What is a myth in your opinion? Which myths are you referring here? Which history are you talking about?

I will assume that you are talking about western view on history or about western history, and that you are asking when did people in western societies started to view religious stories as myths not as a scientific fact. I assume that because you are giving the example of Saint Augustine and Romulus. For this, I would give you the answer that development of science, starting in 17 century had a big impact on defining what is scientifically true. Or in your words, what is not a myth. Again for more specific answer, you should ask questions about specific myth.

But you should not view history as something that is "myth free" even today. That's why I asked what is a myth in your opinion. And sometimes history is really hard to prove scientifically. Nobody today (probably) thinks that Roman and Greek goods are fallen angels. Did we prove it scientifically, that Roman gods are not fallen angels? Also, history is changing. We know something today, but we can discover new thing tomorrow, and suddenly, this first thing we knew for years is now nothing more than a myth. In addition, there is no "one history to rule them all". There are many histories, from different time periods or societies. Something that is a myth for one society, maybe is not myth for another. But most importantly, history can be used as a political tool to prove, for example, how some people are better than the others. Here, you don't need to look at some distant past, you can look at something that happened 83 years ago: WW2. Myth about superiority of one human race was used as a historical fact, and we are still living with consequences of this act. So, if you are talking about religious myths in western societies, development of science is your answer, and again, you should ask about specific myth for more historical answer, but I wouldn't be so sure that we can say that today there is no myths in history.

Mesopotamians fused myths and history in their chronicles. The Greeks didn't, starting from Herodotus and Thucydides (5th c. BCE). By that point, Greek historians were looking for historical answers in human behavior. They put gods and legends aside.

Evhemerus, a 4th c. Greek philosopher, came up with the idea that myths were distorted versions of past facts. He pitched that facts had been altered and exaggerated over time. Zeus wasn't a god, but a king of old, for example.

Many centuries later, Augustine, as many early Christian theologians, were at odds with Greek myths for the fact that they utterly loved pagan literature (it was beautifully written--and much more entertaining than the Bible) but they couldn't condone its content, especially from a moral point of view. Incest here, rape there, murders all around... That was a no-no.

Evhemerus' old thesis was "godgiven", so to speak. It allowed Christian theologians to 'moralize' pagan myths, quite literaly. The Middle Ages saw the rise of an entire genre of literature solely dedicated to the moralisation of pagan myths: how they are actually altered facts, what moral they secretely contained, how they related to the Christian dogma from an allogorical point of view... It was a real fire sale.

The Middle Ages also saw the birth of another literary genre: the hagiography. Hagiographies were the biographies of holy men and women. They were very, very popular. Like ancient legends and myths, they were told "by the campfire" and embellished with some magical elements. By the 16th c. AD, some Jesuits started to question the historical authenticity of hagiographies. I mean... you had saints fighting dragons left and right and more improbable fictitious 'facts' woven into those Christian fairy tales.

Since then, history renewed and enshrined the divorce between myths, or legends, and facts. But for a long time and from the very start, history has been conceived as a literary discipline. To be considered as a great historian you had to write compelling stories. Still today, it means that historians often flirt with that 'edge' between facts, embellished facts and straight up legends.

If you wonder how to evaluate the truth of a claim, keep reading here ;-)