(a|b)(x|y)
{"ax", "ay", "bx", "by"}
While it's true that these other languages have more to them than their syntax, they do define a "language" in the above initial sense: the set of all valid instances of it (i.e. without syntax errors), the set of sequences of symbols.
Programming languages generally include ways of extending their language (in the initial sense). Even java: a java program includes a syntax for extending its syntax (its "language"), in the sense that a program using a certain method invocation becomes valid, if that method is defined. Thus, it is itself both definitions of a grammar, and instances within that grammar - like XML and XSD combined in one (or XML and DTD).
BTW: this reply (and the two similar ones) will probably annoy you, because you know what a "formal language" is (at least, you use the term). I think your misinterpretation is that the article does not claim anything about "natural languages" - only the shape/structure of a language ("So, what shapes languages? Grammars do."/"Behind every language, there is a grammar that determines its structure.").
To be fair though, it then jumps straight into "A grammar defines a language.", without noting a shift in the meaning of the term "language". I think its meaning is clear from context, but it's certainly misleading to shift terminology as you go along!
Similarly, the 'language of languages' is also appropriate given that BNF is defined with a grammar, and is used to specify grammars.
Furthermore, formal grammars were originally invented for purposes of exploring natural languages.
Sorry if you didn't enjoy the article, but there's nothing wrong with it in the context of formal language theory :)
I'm not even sure if a XBNF is the best way to describe or reason about language syntax. Precedence grammars (with hacks to handle braces) are quite interesting for robust error tolerant parsing, and might more closely mirror how we internal grammars in our head.
Grammars are only one part of understanding a language, hardly the "language of languages". In natural languages, grammars are one subset of linguistics. It would be just as valid to say vocabularies or phonology are the language of languages as it would be to say grammars are.
Other than these overly broad arguments and attempts to define natural languages in the same way that formal languages can be defined, this is a nice general introduction to some specific notation techniques for computer languages.
Of course, I might not have read it at all if it were titled "An Introduction to Backus-Naur Form, Extendend BNF, and Augmented BNF Notation Techniques".