How does the clock postulate apply in non-inertial frames? 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?
 A: 
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.

This is indeed the correct general formulation of the clock hypothesis. This formulation applies for all reference frames, inertial or not, and for all spacetimes, flat or curved. The only restriction is that the coordinate basis must have one timelike basis, $dt$.
Because of this, time dilation can be written as $$\frac{1}{\gamma}=\frac{d\tau}{dt}$$ where $\gamma$ is the time dilation factor and $d\tau$ is the proper time, which is related to the metric by $$ds^2=-c^2 d\tau^2=g_{\mu\nu}dx^\mu dx^\nu$$
So, for an inertial frame we have $$ds^2= -c^2 dt^2+ dx^2 + dy^2 + dz^2$$ $$ \frac{d\tau^2}{dt^2}=1-\frac{dx^2}{c^2 dt^2}-\frac{dy^2}{c^2 dt^2}-\frac{dz^2}{c^2 dt^2}$$ $$\frac{1}{\gamma}=\sqrt{1-\frac{v^2}{c^2}}$$ which is the usual familiar time dilation formula.
Note that it has the property that it depends only on the velocity and not the acceleration. When calculating $d\tau/dt$ we get terms like $dx^2/dt^2$, which is the square of a component of velocity, not an acceleration like $d^2x/dt^2$ would be.

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

Although the general formula does work in non-inertial frames, it still does not give a situation where acceleration directly causes time dilation. Let’s work it out for the Rindler metric you mentioned. For convenience I will use units where $c=1$: $$ds^2= -(gx)^2 dt^2+ dx^2+dy^2+dz^2$$ $$\frac{d\tau^2}{dt^2}=(gx)^2-\frac{dx^2}{dt^2}-\frac{dy^2}{dt^2}-\frac{dz^2}{dt^2}$$ $$\frac{1}{\gamma}=\sqrt{(gx)^2-v^2}$$
Now, you might be tempted to say “look, it has $g$ which is a pseudo gravitational acceleration, so the Rindler time dilation is directly related to acceleration”. However, closer inspection shows that it is not just $g$, but $gx$ which is the form of a gravitational potential, not gravitational acceleration.
Furthermore although $g$ has units of acceleration, it is a property of the particular coordinates chosen, not the acceleration of any worldline. The time dilation of a specific object does not depend on that object’s acceleration, only its position and velocity with respect to the chosen coordinates. If an object is momentarily at rest ($v=0$) at some $x=x_0$ then it’s time dilation is $\gamma=1/(gx_0)$, regardless of what that object’s acceleration is. So this time dilation term is a function of position, not acceleration.
It turns out that this is a general fact. In the metric the terms $g_{\mu\nu}$ are functions of position only, not velocity nor acceleration. And so when you divide by $dt^2$ you only get terms that are functions of position times velocities, like $$g_{xy}\frac{dx}{dt}\frac{dy}{dt}$$ You simply cannot get anything else because of the form of the metric.
A: I think one way to resolve your conceptual issue is to consider that motion causes the effect of time dilation, and acceleration causes motion, so that it is true to say that acceleration causes time dilation. However, there are some other points to bear in mind, as follows...
The first is that the effect of time dilation is entirely symmetrical between two inertial reference frames, so if you are time dilated by 80% in my frame, I will be time dilated by 80% in yours. That is because inertial motion itself is an entirely relative effect between two inertial frames.
Acceleration, by contrast, is absolute and thus can lead to asymmetries, as is the case in the twin paradox, in which time dilation is reciprocal during the parts of the travelling twin's journey in which she is coasting inertially but not during the acceleration. Note that it is possible to reframe the thought experiment to eliminate acceleration by having the outbound twin hand-over the baton to an third person travelling inertially back to Earth, in which case all of the asymmetry is associated with the switch of reference frame at the hand-over event.
You can model the time dilation effects of more complicated relative movements between two bodies by summing the time differences over a series of inertial legs with a switch of inertial reference frame between each leg. If you consider that you can increase the accuracy of your model by breaking down the motion into progressively shorter and shorter legs, you will see that in the limit you will eventually replace your summation by an integral.
In summary, the form of motion for which time dilation effects can be calculated in the simplest way is inertial motion, in which case only the relative velocity needs to be taken into account using the normal time dilation formula. For more complicated motions, you must perform some form of summation or integration along the timelines of the objects being considered. During any stages in which the motion is inertial, the time dilation effects are due to velocity and are reciprocal- acceleration introduces asymmetry.
A: 
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?

The postulate is meant for inertial frames only. As you point out in the example, in non-inertial frames, it is not valid. A clock in artificial gravity well (in accelerating rocket) runs slower than a clock that is higher in that well. This is true also in real gravity well, even without any apparent acceleration.

We can ask the same question about two clocks that have no motion relative to each other at all but one is higher than the other in a rocket that accelerates "upward" (or in a gravitational well if the postulate also applies to general relativity). In this case, the lower clock's time is dilated. How is this caused by velocity rather than acceleration? In other words, how is it consistent with this formulation of the clock postulate?

In case of accelerating rocket, the difference in speed of two clocks can be explained in any inertial frame as a result of the standard time dilation; the front of the accelerating rocket is a little slower than the back (due to increasing Lorentz contraction), and this difference makes for the clock speed difference.

Now there seems to be an alternative formulation of the postulate. ...
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.

Only in the sense that different acceleration will result in different velocity and different path in spacetime. The statement on metric is a result of the postulates of special relativity, which do not allow time dilation to depend on acceleration; time dilation in SR is determined by speed in inertial frame.
In reality, hypothetically, clock speed can be influenced by their acceleration, and when this is experimentally found to be a universal effect, not due to some failure of the clock design, then special relativity will be disproven.

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?

Again, this is from a non-inertial frame. Postulates of special relativity are not meant to be valid in non-inertial frames.
A: Let's say bunch of physicists on a rocket try to figure out this problem.
So far they have noticed that

*

*The more they press the gas pedal the more the clock on earth accelerates

*The more they press the gas pedal the more the clock on earth is time dilated

*The more they press the gas pedal the more the frame of rocket is non-inertial

Now it might be so that the increase of time dilation is caused by the increase of acceleration of the clock. Or it might be caused by the increase of the non-inertialness of the rocket frame.
As they are good experimental physicists, they conduct an experiment in which the clock does not accelerate more when the gas pedal is pressed more. They bolt a pre-programmed rocket on to the clock. Said rocket produces such thrust that the acceleration of the clock does not change when the gas pedal is pressed more.
They then notice that the time dilation of the clock still changes, although the change of acceleration of the clock is zero.
So their conclusion is that acceleration of the clock does not have an effect on the clock.
The above mentioned "gas pedal" is a pedal that alters the acceleration of the rocket in an inertial frame. In the rocket frame it is a pedal that alters the amount of non-inertialness of the frame, in other words it's a pseudo-gravity adjustment pedal.
