iPhone automatically deletes every single email on Yahoo server?
All the devices were using the same email account. But every device except the old Macbook accessed it with IMAP, the old Macbook used POP. (I realised this very late.) This caused that the old Macbook downloaded all the emails and deleted them from the server, and that's why the other devices couldn't see the emails.

SOLUTION: Created new IMAP account in Mail, then copying all the emails from the POP account to the IMAP account. The other devices were then automatically able to access all the emails.

(For someone not knowing anything about IMAP or POP, just google the difference between them.)