Correct, though there are multiple interpretations of what a "regex" is. In the theoretical sense you're correct, though some (most programmers, I'd say) mean "PCRE" when they say regex. And PCREs are pretty powerful, see [1] for example

[1] https://news.ycombinator.com/item?id=9748736

Regex has two different meaning depending on the context.

One is CS definition of regular expression matcher for regular languages. The other is extension of that into non-regular languages, typically PCRE compatible.