I apologize up front as this is rather lengthy

You are comparing two days when the first quarter moon occurred, and when the Moon passed over your meridian on that day. You need to find the exact time the first quarter moon occurred.

OK. A little time to discuss this.

Tom is correct here. My original post presented a static view of things when the situation is a dynamic one. Though the original post was accurate with regards to the static case, the dynamics of the situation must be considered.

The original illustration is correct for one specific case; the one in which the previous new moon occurred at 3:00 in the sketch. For a new moon which occurs at 3:00, the 1st quarter moon will appear as shown with the sun leading the 1st quarter moon by approximately 6:12:00 (6:12:30 is more accurate) as it passes the viewers meridian.

So now to the dynamics. The moon lags the sun by 50 minutes every day (approximately 12.2 deg). At this point I'm going to switch to degrees rather than a clock in my discussion. This leads to a time between new moons of 29.5 days. RE and FE models both agree on this point. If we now look at the 2-D geometry of the situation, we'll see this. If we allow the original new moon to be at 0 degrees, with the viewer on the 0 degree meridian, this would mean that the next new moon would occur 29.5 days later which would put the second new moon at 180 degrees.

This would then put the first quarter moon at 0 degrees, sun at 90 degrees for the original 1st quarter and 180 degrees sun at 270 degrees for the second 1st quarter when the 1st quarter moon passes the preceding new moon meridian.

As my original post accurately stated, the time between the first solar noon and 1st quarter moon meridian crossing would be 6:12:30 each occurring at 0 degrees according to the FE model. Now to the second 1st quarter moon. The sun will cross the zero degree line of the viewer after traveling 90 degrees. During this time, the moon will now lag 270 degrees by an additional 12:30. As the sun rotates to 90 degrees the moon will have lost another 12:30. Do the math and the second 1st quarter moon meridian crossing lags solar noon at the point of the viewer by a time in excess of 6:37:30. If we think about the next lunar cycle we would expect the time to return to 6:12:30 as the new moon occurs back at 0 degrees. At this time, I will admit that these times are approximations. However, the observed data fall well outside of any possible error.

At first, I thought it might take awhile to find the proper data points. Turns out I got lucky. The city I've been using as my reference is Portland, OR. As luck would have it, at the 1st quarter moon of Dec. 21, 2020 solar noon leads the lunar meridian crossing by 6:12:00. The max error for this would be +/- 1 minute. If the FE model is correct, this puts Portland at very near the location of the previous new moon. The FE model would then suggest that at the previous new moon of Dec. 14 we should see solar noon and lunar meridian crossing line up +/- 1 minute. Turns out the difference is 10 minutes. Problem 1 with the FE model.

Now let's look at the January 2021 1st quarter moon. We would expect the crossings to differ by 6:37:30. What we find is that they differ by 5:57. Problem 2 with the FE model.

Now let's look at February 2021 1st quarter moon. We would expect the crossings to return to 6:12:00. What we find is that they differ by 5:59.

That's my analysis of the dynamic situation. I fully leave open the possibility that it is flawed. Would not be the 1st time I've made a fool of myself. I look forward to the rebuttals.