Jaime Rudas said:

I consider that those who modify the initially proposed scenario are those who propose that the bubble starts from the Earth.

It's your scenario; it's up to you to specify it the way you want it specified. If you want to specify that the ship starts at some distance ##d## from Earth and ends at some distance ##d## from Alpha Centauri, then just say so. Claims like the one quoted above add nothing to the discussion and are just wasting everyone's time.

Jaime Rudas said:

In the original post I did not categorically claim that the distance from Earth to Alpha Centuari following a path through the Alcubierre bubble was 4.3 light years, but rather I raised it as a doubt.

You ended your OP with the question: "Wouldn't it be like that?" That means you think the answer is yes, it would.

Jaime Rudas said:

it is much smaller than 4.3 light years, but it is not insignificant compared to the distance between the bubble and Alpha Centauri which you claim is much less than 4.3 light years.

First, this is a quibble. Suppose the distance ##d## is 0.1 light year. Then we have a start point 0.1 light year from Earth, and an end point 0.1 light year from Alpha Centauri, and the distance between them, measured outside the bubble, is 4.1 light years. Fine. Then just substitute "start point" for "Earth", "end point" for "Alpha Centauri", and "4.1" for "4.3", and we have the same situation as before. From the viewpoint of the ship inside the bubble, the distance from start point to end point must be much less than 4.1 light years, because the ship takes much less than 4.1 years to cover it and the ship's worldline is timelike.

Second, no, that's not what I claimed. Which you recognize in your very next sentence:

Jaime Rudas said:

Additionally, you calculate the path taken by the bubble to Alpha Centauri, but this is not the same as distance between departure and arrival, because we are dealing with an expanding and contracting space.

What you are really saying here is that there is no such thing as "the" distance between Earth and Alpha Centauri (or between the start point and end point, if we make the change I described above to account for the distance ##d##), because we can specify different paths through spacetime that could be called such a "distance", and they have different lengths.

Which is fine as a starting point for investigation, but if you want to take this approach, then you need to specify which path through spacetime between Earth and Alpha Centauri you are interested in. At what event on Earth's worldline does the path start? At what event on Alpha Centauri's worldline does the path end? At what event does it enter the bubble? At what event does it exit the bubble? Why is the term "distance from Earth to Alpha Centauri, as measured through the bubble" appropriate to describe this path? (Drawing a spacetime diagram can be helpful for addressing these questions--note that there is one, at least a heuristic one, in one of the papers I referenced before.)

The last becomes particularly important when you compare whatever path this is with the one I described, that you call "the path taken by the bubble to Alpha Centauri", and which, from the point of view of the ship inside the bubble, is the natural thing to call "the distance from Earth to Alpha Centauri as measured through the bubble".