My understanding is that there is indeed a difference in what is meant on each side of the pond.
When Brits talk about 'paper piecing' they usually mean when paper templates are cut out, covered in fabric, and then hand-stitched together. Stereotypically this can involve hexagons.
"English paper piecing" - gets shortened (in Britain) to "paper piecing".
When Americans talk about 'paper piecing' they mean using a paper template to stitch fabric onto a paper pattern ('foundation') using a sewing machine as in this YouTube video
https://www.youtube.com/watch?v=H_w35u9gICk (hope that works - if not there are millions of YouTube videos of this!). "Foundation paper piecing" gets shortened to "paper piecing".
Most of the discussion/talk about quilting on the Internet tends to be American-based, so usually when someone talks about paper piecing I assume they mean with a foundation paper and a machine, but it took me quite a while to figure this out!!!
Hope I've explained this clearly...