I looks like alayne is using 'purely functional' to mean Pure + Functional, but you are using it to mean Everything is an Expression / First-class functions, etc.. The functional paradigm.
Lisp is definitely a functional language. It's just not as pure as Haskell, which is the poster child for Maximally Pure FP, if not for the functional style.
My first functional programming language was Caml Light, back in 1996. Followed by Prolog and eventually Lisp.
All alongside traditional lambda calculus and logic proofs, with ocasional references to a programming language called Mirada.
So for me, Lisp is functional programming and I don't buy into this modern notion that only Haskell is the poster child of FP.