The exact text quoted from Wikipedia contains:
"...the moving clocks were expected to age more slowly because of the speed of their travel"
Your question is "Is it really true?"
The answer is "No, it is not true." That is, it is not true that the clocks were expected to age more slowly due to their speed of travel. What is true is that one clock was expected to age more slowly, and the other more quickly. That is not only what was expected, but it is also what was in fact found.
"Ohh, please... OP surely was questioning whether there would be any difference between the clocks, not whether the exact wording in that passing reference was correct."
Maybe. But when you are basically in over your head, trying to make sense of things as foreign as relativity, even small mis-statements like that can really trip you up; it sure trips me up, anyway.
OK, but on to your real question, which I suppose is really this:
Where is the asymmetry coming from that causes one plane to age differently than the other?
You can find a frame of reference in which the experiment is entirely symmetrical; both planes accelerate the same amount, travel the same distance, at the same speed. This frame of reference is not the one that was used for the calculations, but why not? Why is the particular frame of reference where the earth is spinning any more valid than any other? You could even come up with a frame of reference (one that is spinning twice as fast as the earth) in which the asymmetry is reversed. What's so magical about the frame of reference where the earth is spinning at 360 deg/24hr?
And the answer, I eventually came to understand, is that that frame of reference in which the experiment is symmetrical is not an inertial frame of reference, and the one that was used for the calculations is (or is more so, at least).
Rotation is not "relative" it has some absolute meaning (not that I yet understand exactly what that meaning is). The earth is spinning really. For a plane to travel with the earth's spin, it first needs to accelerate. To travel against the earth's spin it first needs to decelerate, in real, not relative terms. Therefore, the experiment is not symmetrical.
To at least begin to understand why rotation is "absolute" in some way, or at least to believe it, if not understand it, think about this:
Consider two balls on the ends of a string. Neglecting the gravity between them, if the balls and string are at rest, or moving uniformly, the string between the balls will be slack. However, if you were to spin the balls about the center of the string, the balls would try to fly apart from each other, and the string would be taut. There is no way to look at that scenario, from any reference frame, where the string is slack.
One other thing: I don't believe this is correct:
"...only acceleration/deceleration takes effect when comparing two clocks in the same frame after certain kinds of motions. ... the time difference of two clocks in the aforementioned test is the same regardless of the length of the flight"
As I understand it, after you are done accelerating, you are in a new inertial frame, with a different time scale. The longer you remain in that new frame, the more difference your clock will accumulate, and will be apparent when you finally return to your original frame.