The illustration was based on the observer always having the traveler under observation. Never, ever, losing sight of the traveler. Physicists have never understood the distortion of their own mapping of the territory.
A traveler traveling away from an observer but remaining under constant observation by the observer -- as the map maps -- would not really be under observation at all. A light born image of him would be under constant observation, not him in the flesh. This continuous light born observation would distort regarding time as lengthening distance lengthened the time it would take light to travel from the traveler to the observer. That is, itself, time stretching in progress.
The traveler, the real traveler and his clock, will gradually get ahead in space and time of the lagging observation of them by the observer. This means the traveler will appear to be slowing down in aging, and his clock will appear to be slowing down in clocking time, distance growing between ticks of the clock. The light born image under constant observation as to physics will be falling behind in time both the time of the real traveler and, at the same time, the time of the observer. It will also be falling behind, not only in time, but in spatial locality. The real traveler will be apparently traveling faster than the speed of light since he will be outrunning and gaining distance on his own image as far as the observer in the rest frame behind doing the observing -- of the light born image -- is concerned. The shift of the light from the growing expansion in distance between is redshift.
This is the true map of the territory.
When and if the traveler reverses course contraction in space and speed up in the movie images will begin and progress. The traveler will appear to be aging faster now, from slowing down in aging before, and the clock will appear to be clocking time faster, also from slowing in clocking time before. Now the image will appear to age faster than the observer rather than slower, the clock to clock faster than the observer's clock. And the shift of the light from the growing contraction in distance between is blueshift.
This is the true map of the territory.
The difference that grew between twins will have disappeared upon reunion, all other things than simple parting and travel being equal.
If time actually slowed down for the traveler and his clock for simply speeding away from the rest frame of the observer, can you imagine what his observation of the observer separating away from him would have been? It would in no way match one for one the reverse observation of him by the observer. Though space and time would be lengthening between them, and light taking longer to travel in each direction between them, all at the same rate, there would be no appearance, to the traveler, of slowing down in aging by the observer, or slowing down in clocking time by the observer's clock. He must see, observe, all of it, most especially the time dilation, happening only to himself and none of it happening to the observer, to Earth, or to the Sun of Earth so rapidly speeding away from him. He must have a sense that the speed of light itself is speeding up, one way only(!), in bringing to him the observer's light born image.
What he must really see is an Earth, and Earth bound observer, falling behind in space and time as fast as he is. He, too, must lose the reality of the observer, the real observer, and the real Earth, then appearing to travel faster than the speed of light, just as the observer will lose the reality of him to the same appearance of traveling faster than the speed of light, as both get ahead of the game of light lengthening and light itself appearing to slow down for lengthening as distances between increase.
But, again, this is based upon all other things, all other physics, being equal, including a constancy of gravs of acceleration being equal.
The above picture matches almost precisely what physicists talk as actually physically happening to travelers. There seems to be no recognition of the fact that there are two of everything, one in the light universe, and the other out front of it in the dark universe. Even the universe observed at 13 billion light years distant is behind the times of whatever is there now by 13 billion years. So the traveler being observed to be a distance of a tenth of a light second distant from any observer is one tenth of a second behind the times of the real traveler, and so is his relative clock a tenth of a second behind the times of the real clock. We are told a traveler can get behind in time, slow down in time, and simultaneously look it in arrived light, which means light would have traveled instantaneously across the distance.
Of course I said "whatever is there now," meaning 13 billion times 6 trillion miles from Earth and the observer, and someone will up and say nothing is there "now." Was there 13 billion years ago, yes. Nothing there "now." You know.... the business of there being one and one only universe, one and one only traveler, the relative universe and traveler in the math.