I have spent a good portion of my career with security clearances. If there is a background check that involves sending actual humans out to perform interviews then be absolutely up front about everything because they are going to discover it anyways. In this context being upfront saves lots of time whereas the opposite could result in additional investigations you don’t want.
If it’s just some mom and pop shop then I fully agree. Just tell them enough to fully and honestly answer their questions without opening additional causes for concern.