If my understanding is correct, back in the 1770s & 1780s, Americans were using "freedom" as something explicitly distinct from their status -- or anyone's status -- as subjects of the British crown. And that status was, at least nominally, much the same as that of present-day British citizens.
However, fast forward to the 20th century, and we see, for example, Winston Churchill, in his 1938 "The Lights are Going Out" speech, framing opposition to Hitler as a fight for freedom. Thus, what he called "freedom" was what the Americans, a century and a half earlier, were rejecting in favor of what they called "freedom".
Something changed in that century and a half. When? And why? (And what other relevant, interesting tidbits of info might there be?)
Surely one difference is that America had "no representation," and were therefore, unlike the electorally represented English, not "free" according to the standards of the time.
I can't comment on the usage rate of the word "freedom" but I think it's generally accepted that the American spin on the concept was in its emphasis on self-sufficiency. Tocqueville's use of "individualism" to describe America was in translation a new word, and one that quickly caught on as something to be proud of (not his intention, by the way). But the philosophical arrogance of the founding fathers in throwing around enlightenment buzzwords irritated the English, who were in their attitudes towards these things rather more circumspect even if they agreed with them in principle.
How is it that we hear the loudest yelps for liberty among the drivers of negroes?
You see in England we take all that for granted.
Liberty was probably the more common word. The French Revolution by turning very strongly against the English tainted liberalism for a long time to come. Shelley's insistence on total freedom got him tagged as some kind of demon. It was controversial to advocate even for tentative moderation on the subject, as society teetered on the edge of a slippery slope leading straight to Bonapartism / Jacobinism. It took a long time—realistically the late 30s-50s—for this fear to subside; for radicalism to emerge from the shadow of the philosophes, for the liberal party to replace the whigs. The first generation romantics disavowed their youthful idealism: Wordsworth and Coleridge retreated inwards and accepted conservative patronage; Southey became a fanatical Tory. The make-up of radicalism in the early 19th century was "fluent Benthamites and muddled Coleridgians." This dichotomy is best seen in the difficult relationship between Carlyle—a very muddled Coleridgian indeed—and Mill, Bentham's testube baby. Carlyle's concept of "freedom" being something tortured, transcendental, and ultimately finding itself only through a madcap devotion to WORK (he was a Kantian). Mill's "on liberty" and the institionalisation of the radicalism of the 30s marked the point where one perhaps became comfortable advocating a peculiarly English kind of "freedom." They however found enough in common to get on in the 30s. Mill wrote a very positive review of "the french revolution." Carlyle, paradoxically, saw Mill's intellectual progress, even in his attempt to escape the shadow of his father (utilitarianism), as the degeneration into a "dryasdust" politicized technician, spiritually barren, slave to the facts. Decidely not free.
On Mill's autobiography,
It is wholly the life of a logic-chopping engine, little more of human in it than if it had been done by a thing of mechanized iron. Autobiography of a steam-engine, perhaps, you may sometimes read it. As a mournful psychical curiosity, but in no other point of view, can it interest anybody.
America is bigger and more diverse than England. England is concentrated and has a much longer tradition. It didn't need freedom as a binding force as much as America did, because America is, in its architecture, an abstract utopia, not a tradition shaped by dense historical evolution. In its rallying cry England would appeal to a shared history not an idea. This is the position Burke lays out in his opposition to the french revolution. Civilisation was the English obsession. Imperialism, free trade, and industry probably helped climatize the English to the benefits of freedom, if ony as a rhetorical device.
However I don't know much about Churchill or the political rhetoric of WWII. But I would imagine the worldwide aspect politics had taken on by that point and his desire to pull America into the fight had no small influence on his diction. The speech you cite is I see directed at the American people.