What you're seeing is the NETBIOS domain name. What you typed in was the DNS domain name. The NETBIOS domain name primarily still exists to provide support for legacy systems, but either are valid.
Active Directory domain names are usually the full Domain Name System (DNS) name of the domain. However, for backward compatibility, each domain also has a pre-Windows 2000 name for use by computers running pre-Windows 2000 operating systems. The pre-Windows 2000 domain name can be used to log on to a Windows Server 2003 domain from computers running pre-Windows 2000 operating systems using the DomainName\UserName format. This same format can also be used to log on to a Windows Server 2003 domain from computers running Windows 2000, Windows XP, or servers running Windows Server 2003. Users can also log on to computers running Windows 2000, Windows XP, or servers running Windows Server 2003 using the user principal name (UPN) associated with their user account.
Microsoft has a support KB article about naming conventions, and valid/reserved names in Active Directory, here.
If you want to log in using the DNS domain name, you do so with the below format:
Where as logging in with the NETBIOS name is done in the following format:
Regarding using "domain.net" or "domain.co.jp," either are valid DNS domain names for active directory, though, since you didn't name your domain with them, they wouldn't work for you. If you wanted to try to use those anyway, you could potentially do so, by setting them up as alternate UPN suffixes for your users.