In the following paper, Chomsky announced the new nomenclature:
In Chomsky (1959)$^1$ a class of grammars is studied each of which
contains a finite number of "rules" of the form $A\rightarrow\phi$, where $A$ is a
single symbol and $\phi$ is not null. Such grammars (there called type 2
grammars) we will now call context-free (CF) phrase structure
grammars.
(Some minor typographical changes from the original, as HSM uses a different greek font for $\phi$.)
The precise detail of the interaction of Chomsky's ideas with Backus's work on defining ALGOL's syntax find the following description in this source:
To address this problem, Backus applied a formalism called
context-free languages that had just been invented by linguist Noam
Chomsky. [..] Chomsky’s work in turn had its roots in Emil Post’s theoretical work on general rewriting
grammars. How Backus came to this synthesis promises to keep
historians busy for some time.
There’s a strange confusion here. I
swore that the idea for studying syntax came from Emil Post because I
had taken a course with Martin Davis at the Lamb Estate [an IBM think
tank on the Hudson].... So I thought if you want to describe
something, just do what Post did. Martin Davis tells me he did not
teach the course until long afterward [1960-61 according to Davis’s
records]. So I don’t know how to account for it. I didn’t know
anything about Chomsky. I was a very ignorant person$^2$.
$^2$ Martin Davis speculates that Richard Goldberg, a Harvard-trained logician and
part of the Fortran team, may have discussed Post's or Chomsky's work with
Backus.
To see the confusion compare also an earlier quote by Backus in Wexelblat, R. L. (ed.) (1978). History of programming languages. Association for Computing Machinery, p. 162 when asked about the influence of formal linguistics:
As to where the idea came from - it came from a class that I took from
Martin Davis. He was giving it at Atlanta State, talking about the
work of Emil Post and the idea of a production. It was only in trying
to describe ALGOL 58 that I realized that there was trouble about
syntax description. It was obvious that Post's productions were just
the thing, and I hastily adapted them to that use. And I'd just like to
add that the main thing was simply the recognition that syntax needed
carefuld escription that really brought it about. It was Peter Naur's
recognition of exactly the same fact independently - I just happen to
have written this funny little paper that he happened to read - that
caused him to use it. But I would say it was a parallel discovery,
namely, that syntax had to be accurately described; that was the real
thing.
There are certainly various folklore histories around these interactions, some giving Chomsky full credit and claiming Backus was aware, some claiming the above independence of discovery as articulated by Backus himself. See for example Cleaveland J.C. and Uzgalis, R., (1977) Grammars for Programming Languages, Elsevier, p.1 for an example of the latter.
I cannot speak to a more detailed historical tracing of changes and adoption of nomenclature, though I suspect that google scholar and google ngram will offer a good first stab pursuing it.
