It's a fair question. If the star was close then it too would apparently move position. If the star was the same distance as the moon, then it would move with it and the relative distance between the two would not change. Since stars are sometimes occulted by the moon (i.e. the moon passes in front of a star) rather than the other way around, we at least know that the stars are further away than the moon.
Now consider two observers at the same latitude. We know that Polaris is less than a degree from due north and that doesn't vary no matter where you observe it from. Similarly for our two observers at the same latitude, Polaris is always the same altitude. What this means is that for these two observers, Polaris is completely fixed in place, no matter how far apart the observers are.
We can then determine the positions of all the other stars relative to Polaris and we find these relative positions are also fixed. The positions of all these fixed objects in the sky are given coordinates analogous to latitude and longitude. These are right ascension (RA) and declination (DEC). You can look these coordinates up in an atlas.
If you want to find some fixed object in the night sky, find an identifiable bright star nearby, point your telescope at it and then alter the telescope settings to match the known RA/DEC coordinates of your bright star. Then point the telescope to the RA/DEC coordinates of the object you are trying to find and if your telescope is properly set up, it should be right there in the viewfinder. This is how we find things in the night sky and demonstrates that the fixed objects are indeed fixed and don't change position no matter where the observer is.
If stars shifted their positions for different observers, then RA/DEC coordinates would vary for each observer and everyone would need their own personalised atlas.
A good way to imagine this is to pretend (note - this is pretend) that there is an invisible, absolutely huge sphere with the earth at the centre. All the stars are nailed to the inside of this sphere and it rotates around an axis once a day. The moon and planets move relative to this sphere, the stars do not.
Since we know the stars' positions are fixed for all observers, the stars provide a fixed background and therefore it has to be the moon whose apparent position has changed and not the star.
Ok....but.... aside from all what you've said, how can you be sure that your star is not a pointed light against your moon say....being only....something like....a few miles in diameter but magnified ?
The great thing about this method is that it doesn't matter what the stars are, what they are made of, how they work. All that matters is how they behave to the observer, i.e. they stay in fixed positions no matter where the observers are or how far apart they are. One observer can set up their telescope, point at the reference star, read off the RA/Dec settings, send them to the other observer and they can use these coordinates to point straight at the same star. The stars make up a fixed background which can be used as a reference to investigate anything in the night sky which isn't fixed.
However if the first observer were to point instead to a specific crater on the moon, read off the RA/Dec coordinates and pass them to the second observer, they would then find that the crater was not in that position. It apparently shifts. You can buy accessories for telescopes which allow you to measure angular distances, so you could measure the shift this way, but all this requires expensive specialist equipment. I'm instead showing you how you can achieve the same result using ordinary consumer grade digital cameras instead.
So are we OK with step 3 now?
How can your star be fixed if you're spinning on your globe at near to or over 1000mph, depending on your position...as we're told?
Suppose you have a roundabout/carousel/merry-go-round - whatever you want to call it. One of those things you find in a children's playground.
Paint a pattern of stars on it.
Have an observer sit on it. Get another observer standing by the trees, some distance away. Now start turning the roundabout very slowly. Once per day. Do the trees start moving? Of course not, they are literally rooted to the spot where they grow. Do the painted stars start moving around, changing position on the roundabout? Of course not.
Things may appear to rotate for both observers, but if each records a timelapse with a 24h interval between frames, then actually nothing moves at all. By all means speed up the roundabout. Doesn't make any difference. Take a timelapse once per revolution and you can see nothing changes. Each painted star remains in the same position on the roundabout, exactly where it was painted. Each tree remains where it was planted.
The stars behave in the same way. Point a camera at the sky. Take a timelapse, with a frame rate of one (sidereal) day. No star moves. They are all fixed.
In addition, no matter where you move to, how far away you move. The stars won't change position. That's all there is to it. They are fixed. Sure they rotate around the two celestial poles, just like the roundabout or the trees rotate for the respective observers, but nothing changes position.
It doesn't matter one bit how far away the stars are, what they are made of, whether they are rotating or whether the earth is rotating. All that matters is they are fixed in place and can be used as a reference for comparison with anything else which isn't - such as the moon or the planets.
OK with this now? Shall we move on to step 4?
If you are still not happy with the idea of stars as a fixed reference point, then I think we have two options.
1) Accept for now what I'm saying, work through the rest of the moon distance method, we note any further issues you are not happy with and then at the end we come back and address all the issues, one by one.
2) Forget about the whole moon distance thing (at least for now) and side-track and start talking about the whole are stars fixed or not issue.
I don't really mind either way. You tell me which direction you want to go in.