How do you return the name of the connection through TSQL? This is visible in SSMS through the properties window. For example, if you have a DNS alias "ProductionServer" for a machine called "SRVR0001" and also have a local server record for this call "Live Production" and then
1) Open a new query that provides a server name of "ProductionServer" and the name of the connection will be "ProductionServer"
2) Right click on the local registry and select a new query, the name of the connection will be "Live Production"
And of course in both cases.
select @@ SERVERNAME will return SRV0001.
This is similar, but different, to these questions:
I anticipate that the answer could be that this is a local property that has SSMS and, therefore, TSQL can not return it in SQL Server.