If you insist on the definition as it is in formal language theory.
In practice regex is widely used to mean the pattern matching thing that also supports back references.
Wikipedia suggests using the term "regular expressions" for the language theory thing and "regex" for the programming language (PCRE) thing. I agree and would even go further and say that any time one wants to refer to the concept as it is used in formal language theory they should explicitly specify that they are talking about the theoretical concept, not the regex implementation as it is found in most programming languages.
If you insist on the definition as it is in formal language theory.
In practice regex is widely used to mean the pattern matching thing that also supports back references.
Wikipedia suggests using the term "regular expressions" for the language theory thing and "regex" for the programming language (PCRE) thing. I agree and would even go further and say that any time one wants to refer to the concept as it is used in formal language theory they should explicitly specify that they are talking about the theoretical concept, not the regex implementation as it is found in most programming languages.