Most people would agree that in functional programming we can pass functions around as first class objects. What we disagree on is the definition of function.
Most languages only have higher-order procedures, not higher-order functions.
As for closures, well, closures are an implementation technique. Not distinguishing between language features and implementation techniques is a part of an established tradition that comes from Lisp, but that doesn't make it any less wrong.
A term does not necessarily become meaningless when it applies to a lot of things. "Functional" might be a broad category after all, not the exclusive name of a subset of functional languages. And if your language allows different paradigms, it will be called "functional and imperative and object-oriented... ". At best, if a property is so common that most language have it, it can be assumed to be satisfied by default. As for "procedural", the definition on wikipedia is a little more precise and does not apply to all languages: https://en.wikipedia.org/wiki/Procedural_programming.
If you can apply the term to every programming language in widespread use today[1], then yes, it is pretty useless. There is real value in having the term "functional programming" be meaningful and denote a certain class of languages; defining that as "any language with first-class function values" is too broad as to render the term meaningless.
It is a bit snarky, but also has a ground truth: pointers to functions aren't functions.
More importantly, you cannot make functions in C. For example, one cannot write a function that, given pointers to functions that compute 1/x and sin(x), returns a pointer to a function that returns 1/sin(x)
That's not a function, that's a closure. And you can simulate that in C by creating a struct that contains the function pointer and the set of captured data (and then when you invoke the function you pass the struct to it as a parameter).