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I don't have a source, but I think it happened between the time of the Eisenhower/Stevenson election and the end of the U.S. involvement in Vietnam. Ike was the first person to call what we now call American conservatism "conservative". At that time, both sides would probably have embraced the word 'liberal', though obviously Stevenson was a socialist.

American liberals through the 1960's were still very pro-defense, and JFK cut taxes to stimulate economic activity very much as Reagan and GWB did.

The recognition that the American Founding (which was the quintessential classical liberal event in all of history) needed to be conserved lead to the rise of American conservatism, which really began in earnest with the Goldwater campaign. The socialists (who could never call themselves what they really were, since Americans mistrusted the idea from the beginning) wrapped themselves in the old word 'liberal', and as the voices of the old version of liberalism in the Democrat Party died out (or became Republicans) the word changed meaning, so that the story has been told since the 1980's of one European explaining to another "in America 'liberal' means socialist".

(Quite frankly, it's a good word, and I want it back. We liberated Red from the left, now I want 'liberal' back. We can call them leftists or fascists, which is what they are really most like now since it's become manifestly clear even to the ChiComs that state ownership and central planning are a dead end.)

I have an analysis which satisfies me quite well:

Socialism is the ideology of denying the significance of any bottom line - it is a fantasy world in which unintended consequences don't interfere with easy "solutions" to complex problems.
Journalism exists to make journalists seem important, and not - as their propaganda suggests - to give us the truth, the whole truth, and nothing but the truth. Accordingly journalism constructs an artificial reality in which journalists are the only people that matter - an artificial reality, that is, in which socialism is sensible.
Journalists and all other celebrities - movie stars, and professors pontificating on topics on which they are not expert, and everyone in between - all toe the line which make journalism look good. Consequently leftism has the prjopaganda power to give it the opportunity to label itself using any nomenclature it finds convenient.
Leftism coined the word "socialism" to obscure the fact that leftism is a tyrannical project, a project of making government co-extensive with society. There is nothing "social" about government power, but leftists talk as if there were no difference between "society" and "government." See, for instance, the tendency of socialists to berate Bush by asserting that "America" contributes to tsunami relief only what the U.S. treasury disburses - completely setting at nothing all private voluntary contributions.
Thus the leftists coined the word "socialism" as a label for what should better be called "governmentism" - tyranny. This was sufficiently misleading labeling for many people in many places, but not in America. Thus "socialism" didn't sell to the majority of Americans, who had seen such betterment of society under libertarian priciples which went under the label of liberalism. Faced with the failure of their brand, American leftists stopped talking about "socialism" - and coopted the label which Americans liked. They simply began using the term "liberalism" to describe what is actually the opposite principle - not liberty but tyranny.
This switch was essentially complete by the time of FDR.