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The earliest form of the quotation seems to be:
I am the civilisation you are fighting for.
It has been attributed variously to Lytton Strachey, to George Bernard Shaw, to the classical scholar H.W. Garrod, and to Lord Dunsany.
(Garrod's Wikipedia article insists that it is correctly attributed to him, but cites a book that relates the anecdote without giving a source. The book in question gives as context Garrod's work in the Ministry of Munitions. But this ministry was not formed until 1915, which, as discussed below, was after the quotation was already in circulation.)
It appeared in print early in the war, in the journal The New Age: A Weekly Review of Politics, Literature and Art. The author of the article attributed the riposte to an unnamed artist:
I heard another good retort of an artist upon a volunteer who reproached him for not enlisting. I, he said, am the civilisation you are fighting for.
[A.B.C., 'Observations and Reflections', The New Age: A Weekly Review of Politics, Literature and Art, vol. 15, no. 16, 20 August 1914, p. 397.]
Since this was only a few weeks into the war, and since many other early appearances of the quotation cite The New Age, one can perhaps conclude that this is its origin.
There is an extensive discussion of early appearances of the quotation and variants in this mailing list post