You run your own server, for a few clients or just for yourself. If mail is sent from that server (you don’t use a relay server) chances are that mail delivery isn’t optimal. Especially to GMail and Outlook/Hotmail addresses.
by martijn
So you updated to Exim 4. Great. Are you using a relay server like Mailgun, Mailjet, Sendgrid, Mandrill, Mailget Bolt, SendPulse or Mailget SMTP? Chances are that emails send to your own domain are not getting through. You will get something like this (Mailgun example):