For me, and I think for most, the more restricted https://en.wikipedia.org/wiki/Programming_language#Definitio... (“A programming language is a notation for writing programs, which are specifications of a computation or algorithm”) is closer to what makes a formal language a programming language.
I think your definition is closer to, if not identical to, what Wikipedia calls “Computer Language” (https://en.wikipedia.org/wiki/Computer_language)
For me, and I think for most, the more restricted https://en.wikipedia.org/wiki/Programming_language#Definitio... (“A programming language is a notation for writing programs, which are specifications of a computation or algorithm”) is closer to what makes a formal language a programming language.