Nicely done.
Nit..."He noticed that the eclipses appeared to lag the most when
Jupiter and Earth were moving away from one another, ..."
As soon as Jupiter moves past opposition, then the Earth is moving away from Jupiter. But it was the extra distance not just the "moving away" that created the time delay for Io's orbit. I doubt he was working on any Doppler-like model.
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"In order to accurately describe the universe, Einstein's elegant equation requires the speed of light to be an immutable constant."
Yes, but it might help some to see that it was the brilliant insight he had that the immutable constant speed of light that was the key to Relativity, which brought forth his E = mc^2.
Because it was becoming clear that motions of galaxies and our planet, and our cluster all would make physics very difficult such that one's inertial frame would require knowledge of all these motions, then something else must be needed. It was getting messy and the top physicists understood this, including Eddington. Fixing c as a constant solved this problem. Einstein called his theory, the Invariant Theory since inertial frames weren't varying after all when it came to the laws of physics. But, this is only my limited understanding of this. A deeper dive would be interesting and important.
It may be worth noting that by fixing c as a constant then, at very high velocities, either the time or the distance to another star system is altered, so much so that at 0.8c, for example, the traveler would cut the time by 40% (or distance). No one knows, AFAIK, whether it is time or distance that is altered. There is much evidence, however, favoring time (atomic clock experiments), and perhaps only one hint of evidence (flattening profile of relativistic particles) that favors distance shrinking.