That definition is decades old.
So defined by whom? On what basis? Can you direct me to the relevant literature? (Don't look too hard, you won't find anything. Since we're playing CS101, you should pay particularly close attention to formal language theory, which you'll find quite illuminating.)
HTML5+CSS3 is known to be Turing complete, yet no one would call HTML5+CSS3 a "programming language". There are other languages which are broadly recognized as "programming languages" which are not Turing complete. (CS101 again: TM's provide just one of many models of computation. Another, which is not Turing complete, would be FSM's. There are many others.)
See, this thread is all about informal definitions. What is considered a programming language or not in a practical sense. The problem, of course, is that everyone here seems to have forgotten that! If you want to be pedantic, a programming language can be Turing complete, or not. They're just ways in which instructions are provided to a computer.