Water meter

Should a home inspection include checking water meter to determine if water may be leaking somewhere on the property

IMO… NO. If you suspect a water leak, call a Plumber.

An inspector I have been training with does it, but only in the process of seeing the bigger picture. It can be tricky. So many things can cause the indicator to turn may not be a leak at all. I also know that you can have a leak that is slow enough that the indicator does not budge, even after 30 seconds or more, so lack of movement is not proof of the absence of a leak.

I think I will probably do 30 seconds of video of the meter with every inspection so I can prove that I at least checked if someone accuses me of not inspecting closely enough.

A turning indicator along with signs up foundation upheaval would be a red flag to me. In fact, any significant foundation shifting is reason enough to be concerned about plumbing leaks, both freshwater and sanitary. Isolated spongy or muddy areas in the absence of recent rainfall, would be worthy of attention as well.