I'll take that as a "yes" and you did check.
When you're looking at the email provider's web interface, the Mac's Mail app is NOT running and a new email arrives…is it read or unread?
I exclude it. It's a personal account, I'm a freelancer and the problem occurs systematically, even with messages just sent and received.'man in the middle' reading your email?
or company monitoring
CC-ing yourself? Seem to recall that does happen in that situation.