On the bottom example, there is too much space between the washer and the wheel. This can cause the wheel to wobble. As far as the washer turning with the arbor, I don't think it matters either way. Sometimes the washer is tight on the arbor and turns with it. It is possible that at some time in the clocks history, the wheel was replaced with a thinner one and now fits rather loosely on the arbor. There should be just enough resistance to turning the arbor without binding.
Yes, that wheel had a lot of wobble and tightening the washer has hopefully solved the problem with the clock stopping intermittently.
The other issue I found was that the pin was poked through the washer, actually punching a hole through the washer rather than riding on top of it. I wish I'd gotten a picture of that.
I've had this clock apart numerous times trying to figure out why it would run fine for days and then only run for 1/2 an hour or so and then stall. As of now it's run for 48 + hours without stopping . I hope the issue is resolved.
Make sure that the washer is fully in contact with the wheel, not just the the end points. If only the points are taking up the slack, they will eventually reveal the gap again. I would suggest you make up a thin washer just enough to bring the washer closer to the wheel at the center. The washer would go on the outside of the original washer closest to the pin.
I hope that Jim DuBois or Peter Nunes can weigh in on this, because they have far more experience with wood movements than I do. However, I'm almost 100% certain that the triangular "washer" is not supposed to be flat against the wheel. The three ends of the triangle are supposed to be the only points of contact. I cannot comment on how much space should ideally be between center of the triangle and the wheel, but the spacing looks right to me.
The 3 legged tension assembly has a bit of dome formed in as originally made. The tips of the tension assembly are intended to force the wood wheel against the arbor. Many of these assemblies carry marks of the tips of the dome washer on the wheel itself. In some instances, the domed washer does not rotate with the arbor. But the pin in the arbor will rotate on the raised portion of the dome.
I did not mean to imply that the entire washer should be in contact with the wheel, but that the ends of the washer should be in contact with the wheel, not just the points. I have seen several clocks that had their gears chewed up by a washer with only the points in the wheel. As far as the gap between the middle of the washer and the wheel, in my opinion, a gap of less than 1mm is my preference.
As a follow up, since I corrected the pin being poked through the friction washer and adjusted the washer so that the arbor spins while the washer remains stationary, the old Seth Thomas has been running reliably since October 3, something it hasn't done in quite some time.