But the differences need not be vague. Everything can easily be derived from the Lorentz Transformation equations, and there is no mystery to it.
If you have an object receding from you at near the speed of light, it will appear to be moving, at a maximum, of 50% of the speed of light.
Why? (Answer this question for yourself, and if you agree, Upvote, if you don't agree, downvote, and post a comment.)
On the other hand, if you have an object approaching you near the speed of light, there is NO LIMIT to it's maximum apparent speed.
Why? (Answer this question for yourself. If you agree, upvote. If you don't agree, downvote, and post a comment.)
Now, when you look at a fast approaching, or fast receding object are you looking at where the object is now? No. You're looking at where the object WAS when it emitted or reflected the light. That emission or reflection of light is an "event" which happened at a place and time in your perceptions. It has physical coordinates of (t,x,y,z) Space and time.
What happens when you accelerate toward a past event in Special Relativity? It moves away from the observer, and back in time. Again, do the math yourself. If you agree, Upvote. If you don't agree, Downvote and post a comment.)
But yes, if you accelerate toward an event in the past, Lorentz Transformation equations say it moves away and back in time. That's good, because it makes everything consistent with what I said earlier:
As the moving twin is moving away from the sun, he's going to see the sun moving away at less that half the speed of light. When he turns around, he's going to see the image of the sun jump away from him--lurching away from him spatially. And it will also (from his perspective) lurch backward in time... So the emission/reflection event happened much further away and longer ago. So at the "instant of acceleration" is when the earth has suddenly aged in his point-of-view.
(Video explanation added, November 7)
The video explains what is meant by "if you have an object receding from you at near the speed of light, it will appear to be moving at a maximum of 50% of the speed of light."
It also explains why the distance traveled by the earth in the second leg of the inbound frame is much greater than the distance traveled by the earth in the first leg of the outbound frame