sql server – SSMS How to find the name of the connection

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:
https://stackoverflow.com/questions/16088151/how-to-find-server-name-of-sql-server-management-studio
https://stackoverflow.com/questions/129861/how-can-i-query-the-name-of-the-current-sql-server-database-instance

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.