Email doesn't reach the inbox

Hi,

I'm having trouble to figure out why my e-mail from say server1 doesn't reach the inbox of a customer that has his mail/mx on another server, but the domain is also on server1 (mine).

I only host the website and dns+mail is external. I have DNS records on my server1 for this domain. I think it auto filled it. Here it says that MX is the head domain, but would the solution be that i change MX DNS record on server1 to the external server to solve this problem? Or should i delete the all but the A-records (for the website) from the server1, because it's only for the website?