So far as I'm aware, from a modern UK point of view, bucket and pail are interchangeable - there is possibly a regional variation in usage, but I'm not aware of it.
Both words come from Old English
- Pail from pægell, which meant "gill" (a small liquid measure - 1/8 pint [c.f.Middle Dutch pegel = measure])

- bucket is from Middle English or Anglo French and is a "container for drawing, holding or carrying water"

So far as English usage is concerned, it probably depends on where you were brought up as to what is right:
As to American usage - I've no idea! They have taken English and preserved some bits we've forgotten and which would by now be lost, and converted some bits which we think would have been well left alone!