Do you have to plant the sweet potatoes first and make sprouts, then cut vines off of these sprouts and put them in water? Can you just cut your potatoes in half and put them into water and grow slips from these?
Well, there are multiple ways of doing it. The ways you described are some of them. What professional growers do in my area is cut them in half lengthwise, and dust the cut with rootone, a rooting hormone powder with fungicide. Then they lay them cut-side down in moist sand and cover in compost in a controlled 70-75 degree Fahrenheit with 80-90 percent humidity. The sweet potatoes will sprout through the compost and root into it and the sand. Then when the sprouts are 6" long, they are removed at the base and wrapped in moist newspaper, letting the leaves out the top, in bundles of twenty five. A rubber band in put around it to old in together. Then it is placed right side up in a small glass jar with an inch of water at the bottom.