How to calculate moon overhead and underfoot time?
At the accuracy you need, the calculation can be quite simple:  It takes 12 hours and 26 minutes for the moon to go from "overhead" to "underfoot", and 12 hours 26 minutes to go from "underfoot to overhead".

So if you notice that the moon is overhead at 20:00 one evening, it will be overhead at 20:52 the next evening,  then 21:44, 22:36 and so on.  (Or, even easier,  approximate to "50 min" and use "add an hour, minus 10 min"

The actual motion of the moon is rather complicated, as the moon's orbit is elliptical and perturbed by the sun. If you need more accurate values you need to use a complex method of calculation, which is what the websites are using.

If you only have rise and setting time, the culmination will be roughly in the middle

The Solunar calculator page contains some obvious errors (such as on 5th May)