Emails inconsistent arriving in both Mail and Thunderbird.
Emails arrive sometimes in both Mail and Thunderbird, sometimes in just Mail, and sometimes in just Thunderbird. Thunderbird is selected email receiver.
iMac Line (2012 and Later)
What I mean is to quit the Mail application. When we quit an app, it is "turned off" and is not working. It is still installed, however. You can quit Mail be selecting it (having an active window) and then press the Command + Q keys. As I said, when the app is closed, it can not and will not send or receive emails.