The term "Regular Expression" is very often misused, it had a very well defined meaning in the field of formal language theory, but like most terms when barrowed by another field some of that meaning is lost or transformed. Some documentation has shifted to using terms like "pattern" or "pattern matching expression" to convey the meaning without the baggage.

> Some REs (i.e. the “Perl-compatible” or “extended” Regular Expressions) are Nondeterministic Finite-state Automata or “NFAs”.

That those engines are implemented using an NFA or a DFA does not actually matter for the question of being regular or not. A given pattern may be Regular while another may not be. There are multiple technical reasons these engines are built on NFA's and not DFA's, supporting non-regular expressions is one, but not the only, reason.

Ironically the library called "PCRE" or "Perl-compatible Regular Expressions" is in-fact not "Perl-compatible" (nor regular). It is at the same time both named "Perl-compatible" and absolutely not Perl-compatible. Both PCRE library and the Perl language have evolved and added mutually incompatible features which results in a valid PCRE matching expression failing to compile in Perl and a valid Perl matching expression failing to compile in PCRE. Just because that is the name doesn't make it true.