I have some sweet potato shoots in water at this very moment. I prised them off the tuber when the shoots were about 4"-8" long, but unlike your example I took no potato at all, just gently bent each shoot back right at the tuber until it came off. Standing in water on the windowsill the shoots have developed dozens of 1" roots and have grown leaves and now look for all the world like true water plants. They have been in water for over 2 months now. As summer is coming to an end here in the southern hemisphere I won't be planting these out. I'll discard them.
I once planted a sweet potato that had some healthy-looking shoots on it. I just planted the whole thing as is. It grew vigorously, spreading to occupy a large area with lush green foliage. In late summer I decided I'd dig just a corner of it to see how big the potatoes underneath were. So I started to dig, and dug and dug, until I eventually had the whole plant up. All I harvested was the same old potato I'd planted! There wasn't a single new one to be found. So sweet potatoes certainly have different requirements from the South American spud.