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Interesting link (above).
I've not heard it as 'the sum sought' in fundraising, nor as in 'it depends what the ask is' but only as ' the asking for, or expectation or hope of, something to be done'. It may have become a noun in two or three different places separately. The US version may have started as the word for the sum sought in fundraising and been applied to other things, as Kerry used it. The Australian version seems to have less specific from the beginning. The UK version seems different again.
A Canadian friend, who lives near the US border, in Ontario, had never heard it used until she returned to England recently, 20 years after leaving. She found it used frequently here and did not think it was American, for if it was general or common in American speech she would have encountered it. (Another new 'anglicism' was 'Bless!' in response to some innocent or naive statement of a child etc)
I like it in the sense it's used here. It condenses a long phrase into one word.