Question:

Why is the Template Label not update when the Template Document Name is updated on the source document in Vault?

Answer:

The source document name is linked to the Template Document field on the Document Template itself and not linked to the Document Template's Label field.

This is working as designed since other documents can be linked to the Document Template while Creating a Document from Template while choosing a template during document classification. This is so the Template Document and the Label fields of the Document Template are not named the same to prevent cross-referencing of fields on the back end. 

Related Documentation:

Vault Document Template Documentation: Configuring Controlled Document Templates