If either K9 or built-in Email are configured using POP3, and Aqua is configured using IMAP (which is Aqua's default for most mail services, as well as auto-discovery), then yes, this is trouble.
POP3 does not have a mechanism to transmit the read/unread message state, or commands to change it on the server.
So most mail services will mark a message read on the server (in web mail, etc.) when it's downloaded over POP3.
Some have a setting in web mail, some don't. Some do it when a message is retrieved completely, some do it even if a mail app only retrieved a message's headers, no content. This is not regulated by POP3, just something everybody does.
Now, IMAP does have message state sync, and Aqua does use.
So what I *think* is happening is:
- You get a new message. Initially its state on the server is unread.
- Aqua syncs the message and its state (IMAP), posts a notification.
- Another mail app syncs the message using POP3. Message automatically gets marked as read on the server.
- Aqua syncs mail again. Oh, this message was unread, and now it's read. Fine, let's sync that. Oh, no unread messages anymore, clear the notification.