Sominemo, 
Haha, new update about the Twitter email delivery issue. Seems like the issue wasn't on my end after all.
I talked to my hoster and they found the issue in their logs: Twitter fucked up HELO header, which triggers anti-spam.
> Postfix: 450 by smtpd through no-relay with code 4.7.1 (<spring-chicken-bh.x.com>: Helo command rejected: Host not found).
Funny that it isn't an issue for Gmail and iCloud, guess these just had to adapt to Twitter's mistakes.