You'd be able to understand my description of Roemer's method, but you won't look past your pre-formed belief that you have the truth, and all the astronomers and physicists are wrong. — Michael Ossipoff
Too bad you have changed your style in favor of polemics. So be it.
Not, it is not a matter of how precise measurements are. In fact, it does not concern measurement at all, except in asking the question:
What are we exactly measuring?
You assume, like everybody else, that it is the speed of light. Once you accept this assumption any inexactitude becomes historically irrelevant. It becomes solely a matter of progress in the measurement process.
I am asking of you to do one step back, before you decide that it is light speed that it is being measured.
When you do that you can ask yourself again:
what am I measuring?
What happens when I observe an event from position p1, and then observe the same event from p2?
If you take the way your observation takes place as irrelevant. That is, if you do not doubt an instant that when you see the event happening, it is happening not at that moment but because light needs time to reach you and make you see it. Then you have already assumed that which you wanted to prove, and the only thing that rests is count the seconds or the minutes between one event and the other.
So, in fact, I am saying that Rømer had already decided for himself that it takes time for events to reach us, and the matter of calculations was then easily resolved.
Apparently, I am unable to convince you that that was an illegitimate jump in his reasoning. That is why we keep going back and forth without anyone of us convincing the other.
For the last time, once you accept the main idea, that is now a given in modern science, then there is no reason to doubt the validity of Rømer's argument.
But you must remember that in his time, that was not obvious and had to be proven. Well, my claim is that he did not, and simply assumed it, then presented calculations and made it sound scientific.
Whether he was proven right afterwards, or not, is irrelevant. What matters is,
was his argumentation valid when he made it?
My answer is NO!