I would say the aspect that defines a functional programming language is the support of higher order functions. I.e. functions that can take functions as arguments and more importantly can return functions as return value.
So I don't think it's very productive to use a definition like this, but not because I want to be divisive, or "move the goal posts" or because I'm trying to be elitist here. It's because when a colleague asks you "what's functional programming?" I think it's much more helpful to describe the kind of programming that is encouraged in Haskell and Clojure than to just say, "well it's just map, filter, and fold in JavaScript".
Same goes for OO, which has something like 9 well-known defining features.
The scenario I described wasn't hypothetical. When a colleague asked our whole office "what's functional programming?" I listened while someone told him that it was just map, filter, and fold. He'd used these in JavaScript already, and was prepared to accept that as an answer. I didn't think that was sufficient, so chimed in and offered some alternative features from other languages that I think are as important as higher order functions, mostly related to minimizing mutable state with managed references and persistent data structures. I think he appreciated my contribution, but I honestly don't know for sure. Maybe it just created more questions.
For example, an instance map is just a cons and a recursive call. No state change needed. You can say something about exposing unwanted details in the cons, but that is a hairier point of contention.
I don't know why, but some years ago, "functional programmers" started to change and twist the definition of functional language step by step, until all those languages fell off and only Haskel remained as "functional enough".
What I do not agree is when "functional programmers" claim that language is not "functional" just because it is not "pure functional" or Haskel alike.
Pure functions. (Only Haskell and friends are strictly pure, as is SQL if you squint.)
Higher order functions. (Java 7 doesn't allow you to pass around a callable function without embedding a link language with reflection or using one-method Runnable.)
First class functions. (Basic, SQL do not have this.)
Generic functions. (C does not have this. C++ does. Java does.)
These are three orthogonal properties a Lang age may have some or all of.
Just not depending on anything in the environment either except globals, which is a huge limitation. So writing a function inc by = \x -> x + by becomes unnecessarily difficult for example.