I’d say there are as many as six kinds of automatics.
In addition to the three named above, the there is the ‘CVT’ as used in the Prius and other hybrids using the same technology. It in no way resembles a CVT as used in other cars, but instead has multiple motors connected to planetary gearsets. The motors can acts as motors or generators, and in some modes one is doing one and the other the opposite. It has the advantage of great mechanical simplicity and reliability, but needs those two motor/generators and some complex control software/hardware. A very neat design that is surprisingly old in concept.
There are still a handful of high end sports cars using automated manuals with a single clutch. Some of the these can shift incredibly quickly, but it’s a brutally abrupt shift compared to what’s possible with a dual-clutch design. Some drivers of exotic sports cars want to feel like Formula 1 drivers, and those transmissions let them do it. That’s about their only advantage.
And finally, ‘conventional’ CVTs have been made with two different basic designs. The more common uses a V-shaped chain/belt (really more of a chain, but it seems more like a belt) running between two wheels of variable diameter (split halves that are moved relative to each other.) I don’t know if the other kind is being used in any current models but it has in the past. It has a disk (or disks) that rides under pressure between two complexly shaped wheels. The disk can be tilted so its edge turns against each wheel closer to its edge or nearer the center, changing the gear ratio. Both designs use some power to force the belt/wheel against the pulleys/wheels, but they are comparatively simple and compact and can have a wide spread of gear ratios. As the belt/pulley design seems to be winning out I’m guessing it has some advantage.
I suspect the CVT will continue to improve in durability and/or serviceability. There aren’t a whole lot of moving parts in them compared to a dual-clutch design and designing a . The ‘motor boating’ effect people dislike where engine speed is sometimes acting opposite vehicle speed can be avoided with a bit of programming or people will just get used to it. The current stair-step rising and falling as you accelerate is no more natural, just familiar. Engines have gotten so quiet I’ve had to point out to my partner when we’re driving a rental with a CVT.