I got 4 email accounts added to Mozilla Thunderbird. I need 3 of them check for new email automatically and 1 only when I press "Get Mail". I went to Tools->Account Settings->my@account.com->Server Settings and removed tick from "Check for new messages every 10 minutes" - but it still gets new email automatically for that account.
Apparently, I also needed to go under Advanced and uncheck "Use IDLE command if the server supports it" option after which emails stopped coming as I wanted.
I found this little piece of information here: https://makandracards.com/makandra/875-disable-automatic-e-mail-checking-in-thunderbird-3