While you’re waiting for an answer to your question, you might be interested in some previous ones in the Graeco-Roman religion section of the FAQ.

For the actual practice of religion, the first thing to keep in mind is that in the Greek world the important thing isn’t necessarily belief so much as cult. Ritual was the most important way of actually participating in religion, regardless of personal belief, and in the Greek world the practice of cult continues even after skeptical philosophical movements start to gain prominence. Even as different intellectual movements come and go, that part doesn’t really change. In his chapter “Philosophical Religion”, Walter Burkert notes that the rise of philosophy in Greece doesn’t change the actual practice of religion in ancient Greece almost at all. All that’s changing is the way some people are talking about it.

On the other hand, this may be one of those places where we can define a thing by its opposite, and if the fifth century saw with philosophy a new rise in rationalism and skepticism toward the gods, that new skepticism was a break from tradition. Burkert goes on to say that:

what does change as soon as philosophy appears on the scene is perspective and verbalization, the kind of questions asked. Previously religion had been defined by forms of behavior and by institutions; now it becomes a matter of the theories and thoughts of individual men who express themselves in writing, in the form of books addressed to a nascent reading public (Burkert 305).

In the 5th century, especially in Athens, which became something of a hub for the sophistic movement, several different not-quite-wholly-distinct rational movements are happening alongside each other. Pre-socratic philosophy took a more and more critical approach toward the existence of anthropomorphic gods (although, important to note, not the existence of the divine itself), alongside a growing corpus of medical writing from which philosophy drew concepts and methodology and the sophistic movement, which emphasized rhetoric, public speaking, and debate, and thus logos rather than muthos.

One really interesting place to see this collision of logos and muthos is in the opening frame to Plato’s Phaedrus. A side note before we begin that the only thing scholars of Plato really agree on is how little they agree on anything, so a full lit review of interpretations of this passage yields some wildly different perspectives; additionally that Plato should be regarded as a hostile source toward the sophists; probably I'm missing several other Plato-related disclaimers but those are the big two. I’ve taken out one part of the exchange that’s not relevant for here, and apologies in advance for the long quotation.

Phaedrus: Tell me, Socrates, wasn’t it from here along the Ilysus Boreas is said to have carried off Orethuia?

Socrates: That’s what they say.

…

Phaedrus: But, by Zeus, Socrates, tell me, do you believe this myth-story (muthologēma) to be true?

Socrates: If I disbelieved it, as the wise men (sophoi) do, it would not be out of place for me, and then, rationalizing it out (sophizomenos), I would say that the wind of Boreas pushed her from the nearby rocks as she played with Pharmakeia, but that it was said of her after she died in this way that she was carried off by Boreas - from here, or from the Areopagus, for this account (logos) is told as well, that she was carried off from there and not from here. As for me, Phaedrus, such explanations are amusing enough, but they are a job for a man I cannot envy at all. He’d have to be far too ingenious and work far too hard - mainly because after that he will have to go on and give a rational account of the form of the Hippocentaurs, and then of the Chimera, and a whole flood of Gorgons and Pegasuses and other monsters, in large numbers and absurd forms, will overwhelm him. (Pl. Phdr. 229b-e)

The exchange here represents some of these new strands of thought floating around in the intellectual soup of the 5th century (the dramatic date of the dialogue, although its compositional date is 4th century). Phaedrus’ use of muthologēma instead of either muthos or logos connotes a skepticism toward the story and aligns him with the schools of rational thinkers willing to look critically at the stories that have been handed down while Socrates, interestingly enough, is here taking the opposite side and pushing back against that (for reasons philosophical can be skipped here since they don't touch on belief/skepticism). This highly critical, rationalizing approach toward myth is very much still something shiny and new in the 5th century and not status quo, but it is out there.

Burkert, Walter. Greek Religion. Translated by John Raffan. Blackwell, 1985.