Thank you very much for your detailed advice and steps to update the global template. I followed them exactly, saved the project file, closed project, then reopened project and the Flag1 field was there again! It seems like something is overriding even the global template.
This is especially concerning since it is across project files. We have another user in our organization who is experiencing the same problem. He also tried the steps to update the global template for the Entry table, and had the same lack of resolution.