I've read countless answers and other sources on the question of whether time dilation is caused both by velocity and acceleration or only by velocity, but they all look at things only from an inertial frame and/or use the popular version of the clock postulate, so they don't seem to answer the questions I pose here. Many of them also talk about it in terms of relative lengths of different paths through spacetime, but I think that sidesteps the question of whether acceleration itself can cause those lengths to change in accelerating frames (e.g., by causing a different metric to apply).
It's stated in numerous sources (e.g., the article "Does a clock's acceleration affect its timing rate?" by Don Koks, posted by John Baez, and this section in Wikipedia) and in answers by reputable users of this site (e.g., here and here) that the clock postulate of special relativity says that time dilation (among other relativistic effects) is caused only by relative velocity and that acceleration itself has no direct effect on it.
But this seems problematic to me because it only ever seems to be true in inertial reference frames. For example, in the (non-inertial) frame of the traveling twin in the twin paradox, the earthbound twin's time slows down during each inertial leg of the trip, but during the turnaround, it speeds up to an extent that indicates that the traveler's acceleration towards the earth is equivalent to a gravitational field that he resists while it pulls the earth towards him thus causing him to experience something equivalent to gravitational time dilation relative to the earth. How can it be said that this kind of time dilation is caused by velocity rather than acceleration?
Edit: To be more explicit about what the clock postulate claims according to some: In a comment, Dale says that "In a non-inertial frame, time dilation depends on velocity and position, not acceleration." The answers given here so far, however, don't seem to support this statement. How can it be supported? Edit: Dale has addressed this in his answer.
Now, there seems to be an alternative formulation of the postulate. According to this paper:
The clock hypothesis of relativity theory equates the proper time experienced by a point particle along a timelike curve with the length of that curve as determined by the metric.
Perhaps someone here will show me that this formulation is logically equivalent to the other one (in all frames, including non-inertial ones), but from what I can tell, it actually allows acceleration to cause time dilation. For example, in the traveling twin's frame where you have to use a metric like the Rindler metric during the turnaround (per the paper, "the restriction to Minkowski spacetime and inertial motion has been dropped"), you find that his curve is shorter than the earthbound twin's during that acceleration and thus that his own time dilates relative to the earth's, apparently due to his acceleration towards it. Therefore, unlike the other version of the postulate, this one works in non-inertial frames and seems to be consistent with acceleration directly causing time dilation. Is this true? If so, is this version of the postulate more correct?