[Turing_Machine]

Don't most DSLs sit atop an existing language, thus giving you easy access to the existing language's ecosystem as well as the syntactic or semantic advantages that led you to create the DSL in the first place?

It's not an either/or relationship, at least not in my experience.

[throwawaymaths]

Yep. A dsl makes it so you don't have to write a new language

[Laaas]

That is an embedded DSL, hence eDSL. A DSL doesn't have to be embedded.