I can't give you a full answer, since Greek hoplite warfare is hardly my speciality.

I do, however, have this very telling anecdote from Xenophon's Hellenica, which Philip Sabin drew attention to in his "Lost Battles."

Then the Argives, filled with overweening confidence on account of their numbers, advanced at once; and they defeated the Sicyonians, and breaking through the stockade pursued them to the sea and there killed many of them. But Pasimachus, the Lacedaemonian [i.e. Spartan] commander of horse, at the head of a few horsemen, when he saw the Sicyonians hard pressed, tied his horses to trees, took from the Sicyonians their shields, and advanced with a volunteer force against the Argives. The Argives, however, seeing the Sigmas upon the shields, did not fear these opponents at all, thinking that they were Sicyonians. Then, as the story goes, Pasimachus said: “By the twin gods, Argives, these Sigmas will deceive you,” and came to close quarters with them; and fighting thus with a few against many he was slain, and likewise others of his party

Take a moment to consider the implications of this incident, reported very matter-of-factly by Xenophon as if it shouldn't be a surprise at all.

The first conclusion is that "hype" played a pretty huge role. The Argives beat the Spartans because they think they're just Sicyonians and thus do not know they should be afraid of them. (This is not spelled out 100% clearly, but this is how Sabin reads it anyway.)

The second conclusion is that the Spartans believe in their own hype. Pasimachus thinks his few men will crush the enemy because they're Spartans. "These sigmas will deceive you!" Only... not so much. He gets his command killed.

Of course, it's very dangerous to draw far-reaching conclusions from one incident (welcome to ancient history, where that's often pretty much all we have to go on) but Xenophon at least seems to report this like it's quite normal for the Spartans to lose when the immense mystique of Spartan Invincibility isn't working in their favour.

And this theme does occur more often in general. You often see in ancient warfare that once a people lose a battle or two against a certain enemy, they start to believe that enemy invincible and will be intimidated much more easily. We see this when the Romans fight Hannibal, or the Cimbri and Teutones. We see it much later when they fight Attila. We see it reflected in ancient battle-speeches. Even if the speeches themselves are fabricated later by the authors, the sentiments expressed therein must have made sense to the people at the time.

Belisarius at Dara, according to Procopius:

You know assuredly that the Persians are not altogether invincible, nor too strong to be killed, having taken their measure in the previous battle; and that, although superior to them in bravery and in strength of body, you were defeated only by reason of being rather heedless of your officers, no one can deny. This thing you now have the opportunity to set right with no trouble.

Conclusion: even if the Spartan reputation was partly hype, even in the ancient world, hype could have a very tangible impact on battles.

Edited for formatting