An affidavit has exactly the same definition in the English legal system, and as far as I can tell throughout the English–speaking world.
Wikipedia mentions Australia, India, Sri Lanka, Ireland, and the United States. Oddly, its only mention of England (and Wales) on this subject is in relation to Ireland, where "affidavits are made in a similar way as to England and Wales".
The word 'affidavit' comes from Mediaeval Latin; it literally means "he (or she) has declared under oath".
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