Based on what we have learned so far, the first example seems to contain a contradiction: According to the astronauts on the rocket, they covered 4.25 lightyears in 3.19 years, I.e., they were going faster than the speed of light.

According to the astronauts on Earth, it takes 4.25 years for the LIGHT to reach Alpha Centauri.

There is no contradiction because, from the ship perspective, the distance to Alpha Centauri is shortened by Lorentz contraction.

Thanks for the advice of dividing and multpling of gamma. I also had a confusion, so I paused the video, First figured out why it got divided and then rememberd this N = YM (N-nomal, Y-gamma, M-moving) . When I resumed the playback, you just solved my problem. Coincidence! And ship reached proxima in 3.19 years due to lorentz contraction.

The Earth moved and came back, while the rocket remained at rest. I don't see how this is not a valid perspective. Otherwise, the Earth gets a preferred frame of reference and that breaks one of the postulates.