Charles in Boston here - a sporadic hobbyist and newbie to the forum.
I've noticed a number of folks using faceting cuts with adjacent torn, roughly-textured sections between them.
My supposition is that the cuts are made and then the remaining clay is left en situ while drying takes place, which is what causes the rough surface when this section is broken off.
I realize the cup in the second image doesn't have the clearly defined chisel marks that Jonathan Cross pieces usually have, and no doubt these greatly help to sharply delineate start and stop points for the tearing. In this case I suspect the cuts were made, but then the cut clay was left to dangle while the still attached end-point dried up some before the tear.
What I'm wondering is if you just let it air-dry, or use a hair-dryer or heat gun in the areas where you want the break? I'm curious if any particular motion of tear or break provides the best texture. Or - and most likely, is this just something that I will need to discover for myself?
Also - the Pottery Park pieces often seem to have an unusual "layered" texture under the tear, but I have no idea of what is causing this - any guesses?
Lastly - while admitting the addictive nature of Pinterest, I'm starting to feel loathing in that its proliferation makes it almost impossible to find the original source of a photo using Google image search.