(Based on what I understand of the quote.)

Classical mechanics (Newton's formula for gravity and Newton's laws for diynamics) is used to describe the motion of the Moon around the Earth.
Describing the motion, means knowing the position and the velocity as a function of time.
Knowing the acceleration of the Moon in its orbit (as a function of time), allows us to predict the motion of the Moon.

In order to be able to describe the motion over centuries, we need observations from the past.
Eclipses are events that have been recorded: knowing the dates of eclipses, means that we have some observations of the position of the Moon on those dates.

Of course, the problem is how accurate those recorded dates are (this is an experimental problem).
If Robert Newton thought the dates were accurate, probably he was mistaken: the fact that he obtained a leap in the acceleration parameter should have made him suspicious about the dates he used.

I cannot comment on the corrections made by Morozov.

Anyway, what is described looks more like this: Newton thought the dates were accurate, but they actually were not.

Though you are not a "math person", I found this review:
Martin C. Gutzwiller, "Moon-Earth-Sun: The oldest three-body problem" , Rev. Mod. Phys. 70, 589–639 (1998),
about the problem of Lunar Motion, from (Sir Isaac) Newton to modern days.


EDIT to add: I have not followed the other thread.

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