Here's my idea:
Thanks for your suggestions on this.
Is that it? Could it be that easy? Or do I overlook/misunderstand a crucial point somewhere?
You also need to edit your SOA record's MNAME field to make ns1.example.com your primary nameserver.
There may be additional steps, depending on your service provider's (GoDaddy, in this case) DNS server.
It's possible their DNS server may refuse to do vanity DNS. (How? By checking your NS/A records and seeing that they are not set to their domain name.) I can't tell you if this is the case or what else you may have to do, as this would vary from service provider to service provider.
I mean, there are some DNS service companies who charge a hell lot of money for this.
I'm surprised at that. Mine doesn't charge any extra fee for vanity DNS.
And: why pay for a 3rd-party DNS solution anyway when you can get the same results for free using your registrar's name servers?
It's debatable whether you can get "the same results".
A third party DNS solution can give you greater reliability, faster DNS resolution, more DNS servers in more locations world-wide, a more convenient management interface, etc. Not all are costly.
When you remove their nameservers why would you remain in their DNS database unless you explicitly ask?
Good point.
I think vanity DNS may not be possible unless the provider explicitly supports it. Most registrars give you a message "your DNS records will be removed within 48 hours, blah, blah" when you change the dns servers to other than theirs.
They may be smart enough to realize that your "new" servers resolve to the same IP addresses as theirs. Or not.