The shadow would certainly produce a local cooling effect. Beyond that, though, I wouldn’t venture to guess what the secondary effects would be. With a system as complex and chaotic as weather, I wouldn’t be surprised if it sometimes created clear spots and sometimes created clouds. All I know is, it’ll have some effect, and any forecast that doesn’t take that effect into account will therefore be unreliable.
Granted, there are no guarantees when it comes to weather even in the absence of an eclipse. However, I don’t think it makes sense to dismiss all weather predictions as unreliable. Also, I assume there are degrees of unreliability.
If predictions are consistently saying that one location is likely to be ~75% cloud covered, and other location is predicted to be ~10% cloud covered, it only makes sense to go to the area with less predicted cloud cover–with the understanding that these are all predictions and probabilities. Even an accurately predicted 10% cloud cover could result in a cloud covering the sun for the 3-4 minutes of totality.
Nevertheless, when making decisions about where to go see the eclipse, all one can do is to use the best information that is available.
In a related note, the Washington Post always includes this caveat in their updated cloud cover forecasts:
That made me laugh! May I ask you a related question, hoping that after 22 responses it is not considered a hijack? The question is: is the limit of the totality called a terminator too? That is: is there such a thing as a totality terminator?
That’s not the usual terminology. Totality has a start point and end point, but I’m not familiar with any special terms for those points.
The region of totality at any given moment is the “umbra”, but I’m not aware of any term for the boundary of the umbra.
Shall we coin one?
A chrona? It is round-ish, so it sounds apt. And it would be a fitting tribute to someone that posts about astronomy in this board.
Umbra, btw, is simply shadow in Latin (if my memory serves me well). Etymologically, umbrellas don’t protect you from the rain, but from the sun.
Flattering, but it sounds too much like “corona”.
Is there a nice term that means something like border or edge in Latin?
What do you mean, “coin one”? Don’t people peruse 18th-century astronomical journals any more? “Limes umbrae, quantum censeo…”
If there’s any cooling effect from the shadow, I’d expect that to increase the probability of clouds. Same air, temperature drop, relative humidity goes up.
Limes, I guess. DPRK seems to agree.
As already noted, microclimates can involve a complicated interplay of multiple factors. In this case, though cooler air will have a higher RH, rising warm air can be an important factor in cloud formation, and the cooling effect of the eclipse inhibits that. Though it’s not likely to have much effect on a heavy cloud cover that’s already there.

why the totality time is shorter at locations near the edge of the umbra than closer to the center line
The moon’s shadow is a cone. The diameter of the cone gets smaller when one is further away from the moon. If the moon is directly overhead, you are 6300 km closer to the moon than someone who watches the eclipse at sunrise/sunset. Some lunar eclipses are hybrid: total for those who see the eclipse at noon (local time), but they are annular for those who see the eclipse at sunrise/sunset.
The Moon’s shadow is a cone, but it can be approximated as a tube of shadow. However, the Earth’s surface is curved. Depending on the season and tilt of the Earth’s axis wrt the sun, the location of “directly under the Sun-Moon line” is where the umbra is circular. As the shadow moves along the curve of the surface, it elongates in the direction of that curvature away from the Sun-Moon line.
Totality is that narrow ellipse that slides along the surface. The speed the Moon moves is constant, but the shadow speed is not. Because the surface slopes away, that changing angle from the shadow axis makes the shadow move away faster at Sharper angles.
For the OP, there won’t be a noticeable effect of the experience, but it will affect the duration of totality and, as mentioned, ability to see Bailey’s beads. But some of that is also how far off the centerline you are.

The Moon’s shadow is a cone, but it can be approximated as a tube of shadow.
I would not put it that way, because it is misleading: the tip of the cone during an eclipse is always in the vicinity of the surface of the Earth. When it ends above the Earth’s surface, you get your annular eclipses and annular-total eclipses [the latter when the surface moves in and/or out of the shadow during the eclipse].
For people within the path of totality but not on the centerline the big driver of the differing duration is simple plane geometry.
The shadow spot on the surface is an ellipse or a near circle. If you’re on the centerline as the shadow goes by you you’ll traverse a line from one edge to the center and out to the other edge. So the full diameter. Which gives the longest possible duration at that place along the totality path.
If you’re out near the edge of the totality swath the passing shadow circle will go by you such that you enter it at say the 1 o’clock position and exit again at 11 o’clock. Having traveled a much shorter distance through the shadow area.
The same idea applies if the eclipse is at a low elevation from your POV. The shadow is much more elliptical, but again the difference is whether you pass it or it passes you across the fat part, or instead you just dip briefly into the edge and back out.

When it ends above the Earth’s surface, you get your annular eclipses and annular-total eclipses [the latter when the surface moves in and/or out of the shadow during the eclipse].
Good point. I stand corrected.
Note, by the way, that it seems like a coincidence that the apparent diameters of the discs of the Sun and the Moon are right around the same like that, because you can imagine a planet with a somewhat bigger if not huge moon, or, as was discussed in another thread, you can wait around even on Earth for another billion years or so and the Moon will migrate outwards, screwing up the alignment in the other direction (no more total eclipses)
For example, Charon as seen from Pluto would be much, much bigger than the sun.
On the other end, Phobos and Deimos do not totally eclipse the Sun as seen from Mars.
For a discussion of an example of an eclipsing body visually much larger than the Sun see here from yesterday:
If a human were on the ground (at the right spot) on the Moon when the Earth eclipsed the Sun, would they see anything particularly spectacular? Or would it just look like a sort of hole in space where the Sun used to be?