OK, got it.
So I have 2 accounts: 1 POP3, 1 IMAP. IMAP push was enabled in the global settings, but not on the accounts. Only when also enabling IMAP push on _both_ accounts, the status bar icon indeed stays there.
But there are 2 things that are not intuitive at all:
-I have to enable IMAP push on all accounts in order for the setting to be really active
-I also have to enable "IMAP push" on a POP3 account - but it also works there, new emails are registered instantly.
I also tried to end the Aqua Mail task with a task manager when in "IMAP push" mode, and it cannot be done until this setting is disabled (this is the expected behaviour and perfectly fine).
So in reality "IMAP push" seems to really mean "Stay in the background and watch for new mails".
Now, what I would whish is a quick setting to disable "IMAP push", like an "Exit" button in the main window to quickly stop the background program...
That would make Aqua Mail the ultimate mailer for me (I'm still using the free version but probably not for long).