EDIT: I opened this topic indirectly encouraged by @davidep .
He commented my answer regarding the name for the internal domain ( Mail bounced (user unknow in virtual mailbox table) ), considering articles that I mentioned, very useful.
In his comment, he summarized very well what was important to remember, quoting from these articles:
"Because the DNS names of all the nodes that require name resolution include the Internet DNS domain name for the organization, choose an Internet DNS domain name that is short and easy to remember. Because DNS is hierarchical, DNS domain names grow when you add subdomains to your organization. Short domain names make the computer names easy to remember.
If the organization has an Internet presence, use names that are relative to the registered Internet DNS domain name. For example, if you have registered the Internet DNS domain name contoso.com, use a DNS domain name such as corp.contoso.com for the intranet domain name.come" (1)
“For your Active Directory Domain Name, use a subdomain of your public domain.” (2)
Here are the links to that articles.
About the same problem, I added two links to articles that, in my opinion, are relevant also.