The nature of eclipses wasn't wasn't specialist or esoteric knowledge. Ancient Romans were aware, as we are, that eclipses are caused by the the earth/moon passing in between the sun and the moon/earth.

Here for example is Pliny the Elder discussing this as evidence of the relative sizes of the earth, sun, and moon, in Natural history 2.51; soon afterwards at 2.53 he reports that the first Roman to describe eclipses in this way was Sulpicius Gallus, when he was a military tribune in the 160s BCE, and claims that the first person anywhere to explain eclipses in this way was Thales, in the early 500s BCE (the bits about Thales are at least partly fictitious, though). At 2.57 he explicitly describes a lunar eclipse as 'when the shadow of the earth causes the moon to be dimmed' (cum conveniat umbra terrae lunam hebetari).

This doesn't mean natural phenomena eclipses didn't also serve as omens: Pliny goes on to talk about instances of that too. But the physical nature of what was going on in an eclipse was well understood. Pliny also draws attention to the fact that lunar eclipses are observed at different hours by observers at different longitudes, and that this is caused by the earth's spherical shape (2.180; the same point is also cited by the astronomers Ptolemy (Almagest 1.4) and Kleomedes (On the heavens p. 76 ll. 11-16 ed. Ziegler) as part of their catalogue of evidence for the shape of the earth.

I don't know where you got your information that the Greeks 'believed eclipses were punishments', but if that claim was accurate in context, it must have been a specific religious or mythological context. Aside from Pliny's unreliable mention of Thales as the discoverer of the actual nature of eclipses (a story that appears elsewhere too), we do have more reliable testimony that Anaxagoras (59 A 42 Diels-Kranz) and Empedokles (31 A 59 Diels-Kranz) correctly explained eclipses as caused by the earth/moon passing between the sun and the moon/earth; Aristotle cites eclipses as evidence of the earth's shape, too (On the sky 297b, 23-30), though in his case it's the shape of the earth's shadow on the moon that points to a spherical shape.

And as I said earlier, the fact that the physical nature of natural phenomena could be correctly described doesn't mean they didn't also get interpreted as omens. The Historia Augusta (4th/5th cent. CE?), for example, claims that a solar eclipse was a portent that the reign of the emperor Gordian would be short (Gordian 23.2); Plutarch (2nd cent. CE) claims that when Nikias was attacking Syracuse in 413 BCE, there was a lunar eclipse which caused superstitious fear (Nikias 23.1). Plutarch goes on to claim that eclipses were enshrouded in superstition until the time of Plato -- but Plutarch isn't reliable about that kind of thing.