Brian, look for which is the Default unit. In your first screen shot, I think I am seeing Foot as the default unit in your project.
Adjust that unit's decimal precision up.
This was counterintuitive to me when I first encountered it, because like you, the dataset I was working with was also in meters, the measures were calibrated to miles, but the default display unit was Foot.
P.S. I tested switching my default units to Miles, and the Explore/Identify started using that decimal precision for distances.