WHOOOOHOOO! It works!
The SMTP Mailer plugin works like a charm. I had to set it up a bit different than your comment @noah, but it works!
THe changes I made: I had to set SMTP authentication to ‘True’ (what is the point of filling in the SMTP username and password if set to false?).
Type of encryption: TLS
SMTP Port: 587 (because TLS)
SMTP Host: not my primary domainname but my one from my hosting provider (mail.<hostingprovider>.nl).
Thanks a lot for your help guys!