To run them as stand alone apps, they must either go though a different interpreter than usual or be emulated. So technically they would be completely different subjects since they are not being ran though the same parser. Regardless, I am not referring to them in that context, I am referring to them as they are normally used.
MySQL isn't. It's an application. Is Visual Studio a programming language? Or PostgreSQL? Recursion panic! I said SQL was a data manipulation language. SQL in a database is also a data specification language and then with stored procedures and flow control constructs it becomes a programming language, as you say.
But the language (t-SQL) that you query it in is commonly referred to as the application's name (MySQL). Its a case of common references. When I say I program MySQL, I am not referring the the application itself, rather the querying language it uses. You just did the same with "I said SQL was a data manipulation language".
An operating system is a program, just one that works directly with the machine.