There is no direct way to do that as far as I am aware.
As a practical matter, we'd sometimes create two logical Stored Processes that can be powered by the same .sas script and define one of the STP parameters (macro variables) to be a constant, different between two instances. Then one business context has one Stored Process leveraged as a Custom Diagram Tool on a panel, and another business context has another. So, you are duplicating metadata definitions, but you can avoid duplicating the actual program code (.sas).
If you are dead set on "getting a Business Context name" from inside the stored process (process node), a possible dirty workaround can be to define different MATable libraries for every business context. You can then use SAS BASE language facilities (DICTIONARY.LIBNAMES) to get the information on details of how MATables libname is assigned in the current session (what path or options, etc), and deduct what was the business context based on that. This is a really dirty hack that you should not be doing unless there is absolutely no other way. I wonder that a use case for that may be.
Hope this helps.
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Dmitriy Alergant, Tier One Analytics