There are many similar questions on here but nothing that looks like it's actually what I need.

I know what I need is very simple but I can't websearch it, probably I'm missing a key word.

My issue:
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There is a standard DNS set up on a third party (as yet unseen by my eyes) controlling the website and email. Defined as "Server Old". There is a new website set up on its own server, defined "Server New" this server hosts the website ONLY and so I need to update the DNS of the original "server Old" DNS hoster.

I understand I should update the original `A` record (only) to the IP of the server hosting the new website.

But I am having trouble understanding what to do about the DNS for the new-website server, it needs to recognise when dealing with email to send emils out to the WAN rather than dealing with them internally. How do I set up *this* DNS, for example if the new website sends emails to the website owner, the email needs to be send from the new server to the old server;

Is it simply a case of updating the MX record on the new-website server to point back to the original DNS server?

Or have I missed some things here?

Update 1
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> 
> In response to [Tilman's answer](https://serverfault.com/a/1112869/278183) to my above query. A kinda rewrite and clarification response to the above answer.
> 
> 
> 

There is a Domain; "Domain Alpha" that has it's A, MX and all other records set correctly.

1. I believe that, in order to redirect the website -*only*- to a new server, that on the DNS rules set for "Domain Alpha" I should update the "A" DNS rule -*only*- to the IP of the new website server.
2. I would then have a DNS ruleset on this new server that would -*only*- handle incoming HTTP requests and anything else be pushed back to the original FQDN DNS ruleset.
3. **Basically, I want to delegate all services EXCEPT email (MX) to the new server and leave the email as it is.**

Can I do this (3) by setting an A ruleset record in the existing DNS ruleset that points to the new server, and then on the new server (which also contains DNS rulesets) setting an explicit MX record back to the "original" current server IP (or the orginal / current DNS host)?