There are so, so many ways to make a split, depending on the time of year, the goal of the split, and the equipment you have at your disposal. I'm bad with the terminology surrounding different types of splits, because there are just so many ways to do it, and I prefer to just think through the process logically each time. The thing to ask yourself when you split a hive is this: what is each half now lacking, and how do I make sure they can manage well until they have it again?
Here is what I typically do, but again, there are 100 ways to skin this cat. When I split for increase, I try to mimic a swarm, so I'll move the frame with the queen into the new hive. I make sure there are drawn blanks for her to lay in, some capped brood, and plenty of stores, because this side of the split will be lacking foragers, and therefore won't have food coming in until the bees that emerge in the new hive begin foraging (which they will do early, in the absence of older bees). All the current flying bees that I have incidentally moved are oriented to the original colony's location, and when they leave the new hive to go forage, they will return back to the old hive.
In the original location, they are now lacking a queen, so in order to make one they will need frames with eggs and also plenty of nurse bees, so I ensure the original colony also has capped brood, since nursing is the job of the young bees. If I'm expecting a big flow, I'll also be sure the original colony has the supers they need, because I don't want to disturb that colony until the new queen has mated.
In your case, Terri, the question is going to be if the hive is really big enough to split down the middle yet. If not, making a nuc off of the hive may be better, but that nuc will have gotten a very late start. Or, you could do something like Phillip described, and wait and make a nuc off your hive later in the season, if you don't catch a swarm. You will then have to overwinter that nuc, but one strong colony and one nuc might be preferable to 2 weaker colonies going into winter. It's just a matter of preference and proper timing in your situation.
