In Wikipedia Domain Specific Language is defined as:

a programming language or specification
language dedicated to a particular problem
domain, a particular problem representation
technique, and/or a particular solution technique

I have been looking for a better "frame", I may have found one.

I was looking at A domain-specific language for experimental
game theory
paper. And this Tutorial in game theory had a sentence that best captured the "DSL think" that I am looking for. On 2nd slide it says:

"To analyze strategic behavior we need the conceptual tool of game theory..."

A DSL is essentially allows you to apply the conceptual tools to your domain. The tools in turn need the underlying data in format that is also covered in your DSL.