I wonder if anyone back then, say an armchair astronomer, tried to argue that it was due to the atmosphere, and backed their argument up by comparing the atmosphere to how a pool of water bends light.
We'll never know.
The refraction is a larger effect, but even an airless planet could have an observer seeing the centers of both the Sun and Moon at eclipse time just above the horizon.
I’d also question if this can have any meaningful effect on the sun, which is about 23500 Earth radii away. That’s a very, very slight perspective shift.
A huge distance is actually a helping factor in this case. A 1.75 meter person can see 4,700 away before the horizon curves out of view. That means (ignoring the bending of light by the atmosphere still) the viewing angle allows for 1 meter "down" every ~2,685.7 meters out. The radius of the Earth is ~6,371,000 meters so to be able to see the middle of an object from ~the radius of the earth "down" with that viewing angle would require the object to be a minimum of ~2,685.7 * 6,371,000 = 17,110,594,700 meters away. That's a minimum distance of ~0.11 AU so the person would only get a partial view of the moon but a full view of the sun because the moon -isn't- far enough away! Well technically the Earth could rotate slightly farther while having the middle of the sun still visible but I'm too lazy to calculate if it's actually enough, my hunch is no. If you wanted to see the middle of the moon and the middle of the sun (with no bending by the atmosphere) during an eclipse you'd have to be near the top of Everest while it happens to be exactly rotated between each.
Forgive the hasty diagram but https://imgur.com/a/aoZIZTs
The Earth is entirely in-between the wildly out of scale Sun and the Moon, but a giant standing on the "top" of the Earth can see both.
This actually wasn't the only time when the refraction of the Earth's atmosphere produced surprising observations in the ancient world. One of the devices the Greeks used to determine the time of an equinox was what is called an equatorial ring. It's basically just a metal hoop oriented at an angle equal to 90 degrees minus the latitude. (Here's a diagram: https://en.wikipedia.org/wiki/Equatorial_ring#/media/File:Eq...) During an equinox the Sun would be right on the celestial equator, and the shadow cast by the upper half of the hoop would fall exactly on the bottom half.
But it was noticed that sometimes, the shadow would fall on the bottom half early in the morning, then cross to the other side, cross back, and then later cross over once again --- it seemed that there was three equinoxes! Ptolemy explained this away as being due to instabilities in the device's orientation --- somehow it was very slightly tilting and so it wasn't possible to measure the time of the equinox to better than half a day or so. But today we know that this was also due to the refraction of the atmosphere. Early in the morning, the Sun would be refracted to appear to be just above the celestial equator, but as it rose, the refraction would lessen and it would fall back to its original place. Then later in the day it would cross the celestial equator "for real" when the equinox actually occurred. So the device was much more accurate than ancient astronomers gave it credit for!