You've answered your own question there
However, what you wrote about isn't strictly time dilation. If you wanted to observe the effects of time dilation, you would need to carry a clock with you and have it synchronised to a clock I had sitting on my desk at home (or in the lab) on the day you left for Proxima. What you would find is that the closer to the speed of light you traveled at, the more out of sync with the clock back in the lab you'd become. A second to you, would still be the same length of time to me as well. But because you're moving much faster than I am, the actual amount of time that you would experience traveling to Proxima would be less than what I would experience here at the lab. To me, your clock would appear to have slowed down. To you, the 4 light years would seem to be covered in a matter of weeks (at, say 0.99c), or even instantaneously at precisely c, however, here at the lab something like 4.5 years would've passed by. The faster you traveled, the less time it would appear to take for you to get there, but for me it would take whatever time it did...whether that was 4 years, 5 years or more (depending on your velocity). However, the slower you traveled, the more in sync our clocks would be because time dilation only becomes considerable at substantial fractions of c. At, say, 0.5c the differences between our clocks might only be about a month or so.
That is the whole idea about Special Relativity...that time, velocity and location of any object within spacetime are all relative. It depends on who is making the observations of the objects in question and what is their position/velocity/time in relation to the objects. There is no absolute fixed point of reference within spacetime.