I know the following question is sure to cause howls of anguish. I can only apologize for the length at which I have had to write in order to be precise, and I do hope people will bear with me for a question asked in good faith.
The following statements are made on the entry for Time Dilation:
The dilemma posed by the paradox, however, can be explained by the fact that one of twins must accelerate while the other remains inertial.
the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity
And, on the Twin Paradox page, the following is asserted:
one must understand that in special relativity there is no concept of absolute present
and also the following scenario is described:
If, instead of incorporating Einstein's clock synchronization (lattice of clocks), the astronaut (outgoing and incoming) and the Earth-based party regularly update each other on the status of their clocks by way of sending radio signals (which travel at light speed), then all parties will note an incremental buildup of asymmetry in time-keeping, beginning at the "turn around" point. Prior to the "turn around", each party regards the other party's clock to be recording time differently from his own, but the noted difference is symmetrical between the two parties. After the "turn around", the noted differences are not symmetrical, and the asymmetry grows incrementally until the two parties are reunited. Upon finally reuniting, this asymmetry can be seen in the actual difference showing on the two reunited clocks.
Firstly, I believe I'm right in saying that, as far as time dilation is concerned, the quantity of difference between the clocks when the twins reunite is purely a function of the speed reached during the travelling twin's journey, and the distance of the journey undertaken at that speed.
Acceleration forces play no fundamental part in accounting for the difference. I would only note that, collaterally, one must accelerate in order to change speed in the first place. But it is not the force of acceleration that causes the difference to be incurred, and the differences are not incurred purely during the periods of acceleration.
That is, the difference accrues more or less steadily over the entire round-trip journey (as proven by back-and-forth signals in the scenario in which each twin examines the state of the other's clock, in which differences accrue steadily), and it is accruing even when the travelling twin in steady motion and no acceleration is occurring at that time.
Were it otherwise, a very short journey incurring given acceleration forces and reaching a given top speed, would have a dilation equivalent to a very long journey involving the same acceleration and same top speed - but this (I think it will be agreed) is not the case.
But again, collaterally, acceleration is relevant in that it provides an objective reason for treating one twin as non-inertial, and it would act as an observable cue to the travelling twin that he has "changed frames" during the course of the journey. But that change, at the moment of happening, is not what accounts for the difference. The difference is on account of motion.
Secondly, I find the assertion that there is "no absolute present" problematic. I differentiate this from the notion of a "relative passage of time", or a "relative clock rate" against which the passage of time is measured, with which I do not take issue.
It must surely be the case that, when the twins are reunited, they are both in a shared moment of "the present" or "now". So too, they were in the shared "present" before the journey began. It also seems reasonable to say that, from the perspective of either twin, the other always remained in the present moment - there was never a moment during which the one twin somehow disappeared from or ceased to exist in the present of the other twin.
The present moment is not itself a quantum of time that is (or can be) measured on any scale - not being a quantity of anything, it is universal in it's meaning and free of measurement. To talk of the present instant is to talk of a static snapshot of the state of the universe, free from consideration of its dynamics or what will be its future states, or any notion of the "passage of time". I think that captures what people would mean by reference to "the present".
It would not seem like a credible composition of the concepts involved to say that, in some sense or another, the travelling twin remains somewhat more "in the past" than the home-twin (though undoubtedly the travelling twin is looking younger), or that the present of the travelling twin is somehow behind the present of the home-twin. He looks younger but he's not in the past. This is because both are clearly together in the same present. So it does not seem to me that a "relativity of the present" accounts for the difference. Certainly, taken as a bald statement without more reasoning, it has no power to communicate an explanation for the phenomena observed in relativity.
Thirdly and finally, assuming the scenario described at the outset to be correct, I'm not clear why the "asymmetry" in the clock rates accrues only on the inbound stretch of the journey. It is said that, on the outbound stretch, that each twin measures the other clock to be running slow, but both measure the other to be slow by equal amounts, whereas on the inbound trip after the "turn-around" this ceases to be the case.
Now to conclude, I don't think I'd be unreasonable in saying that the explanations given on those Wikipedia pages present and communicate a confusing picture for anyone trying, in good faith, to get properly to grips with relativity and gain an understanding. It presents a confusing account of what role acceleration plays in the explanation, it invokes a "relativity of the present" which is a confusing concept when stated baldly, and it describes a scenario with confusing asymmetries that arise on only the return leg of the journey and don't seem to be accounted for by the process being described.
I'd be tempted to cast my question quite widely in terms of "have I missed something?", but to be more specific, what is the explanation for the asymmetry that arises on the return leg? Does it occur specifically on the second leg of the journey, or does it occur specifically on the inbound leg** of the journey? And in either case, what is the explanation for why the difference arises only during one leg and not during both?
**I say "inbound leg", because it would be easy to adapt the scenario where two twins start off on different planets with synchronised clocks, then one flies in to meet the other, and then returns back to the planet where he started (at which point, they compare the current state of their clocks by signalling to each other between the planets). In such a case, would the asymmetry occur up-front on the inbound leg, or would it occur on the second, out-bound leg?