No matter what speed the earth/proxima frame passed by you, your watch would say you did the 4 ly trip in 3 yrs while the earth watch would say earth went nowhere relative to the earth/proxima frame. Relativity defines frames as points. So the only two frames are you and earth. But if you define a common frame that includes both start and end points and the relative velocities between the two point frames are defined through relative velocities to the common frame, you can re-define the word "moving" as distance moved according to a local watch, (instead of the other guy's watch). Therefore Earth's Yv is 0/3 and your Yv is 4/3c relative to each other through the common earth/proxima frame. Who's really moving in some universal sense doesn't matter and is superceded by the distance/local time metric which comes up with an unambiguous answer. There is no way the earth/proxima motion would come up with a relative velocity to the earth/proxima frame as witnessed on their local watch.
PS. Let's say you're hoverboarding a marathon and an asteroid hits the earth causing the finish line to hit you. The relative velocity is not between you and the finish line but between you and the race course start to finish. The finish line's relative velocity to the race course is unaffected by the increased spin of the earth but your relative velocity to the race course is. Sure the earth spun underneath you during the race but your watch says you alone got the boost in relative velocity.
This is no longer a scenario of constant relative velocity, though, unless the asteroid struck while you were hovering at the start line. If the asteroid had spun the earth in the opposite direction and the start line hit you from the rear halfway through the race, this would now be an example of the reverse twin paradox (Bob takes off from earth to catch up with Alice). Since the earth initiated the change in relative velocity, it's clock would end up having aged less than yours and would therefore establish the entire race course is also moving relative to where it was when the asteroid struck.
Edited by ralfcis, 19 September 2019 - 06:22 PM.