There is something wrong. If the traveller when return on Earth is younger than his twin, we are forced to say that the time ran faster on Earth. If the time on Earth had run slower as you said, the traveller would have returned elder than his twin, and for this reason I disagreed (and now too) with you. The example of the clocks can't exist if the time and the clocks are the same thing...
You're on the right trail but you have a ways to go, IMO.
> We agree that on Earth
our clocks won't speed up or slow down in rate. All will seem normal for those on Earth with their own clocks.
> The
spaceship time to reach their destination will not suggest their clocks ran slower. IOW, at, say, a ship's speed of 80% the speed of light, Earth would determine that it took a spaceship about 5.4 years to reach proxima Centauri. [That's 5.4 years using Earth clocks.]
> The Earth observers will see, perhaps to their surprise, that the space ship clock ran slower even after the time delay due to the speed of light is taken into consideration. [Earth clocks say all is normal in just the time it took them to go to another location, but if they (space travelers) report their time of arrival (ie their on-board clock) then we would be told the spaceship clock ran slower (2.6 years of travel time by the spaceship clock - ignoring all the extra time to slow down to get there).
> Thus, when the spaceship returned the same would be true and the space travelers would have aged at a less rate than those on Earth.
> But, and this is where I struggle, the spaceship travelers would argue that the clocks on Earth would not only appear slower (due to the time for light to reach them) but also because of Relativity. Namely, that relativity means that the spaceship could be regarded as still and the Earth is moving away from them. Hence they would determine that people on Earth would be aging slower, at least up to the point the spaceship began its deceleration upon arrival to proxima Centauri, or wherever.
This would make it a true paradox -- since, upon their return, we know the end result of their clock having run slower -- if this was all there was to the Twin Paradox story, but the spaceship had to decelerate upon arrival at the destination and then accelerate to return. If we make this simple relativity, then hitting the engine switch to move the entire universe while the spaceship stayed still is very illogical. [This would explain why the Earth clocks would appear by space travelers to run slower.] Somewhere in that illogical circumstance is where physicists describe the acceleration experience as causing a symmetry break -- where one clock becomes the slower running clock, namely the spaceship clock.
Perhaps a physicist is around to help us on this, especially since I could still have this wrong, admittedly.