I'm no master I didn't even know what a three speed hub looked like till I saw one and you told me what it was. I knew there were three speeds but I assumed they had external sprockets, not internal gears.
Well, I found this on the late great Sheldon Brow's website-
A two speed coaster brake hub, which switches between its two gears each time the brake is applied. The advantage of this system is that it doesn't require any hand controls or cables. They were formerly made by Bendix and Sachs, and were popular on folding bicycles and inexpensive tandems, but are not currently in production.
Bendix (U.S. company) made 3 models, distinguished by colored stripes running around middle of the hub shell.
The "blue band" model offered direct drive plus a higher gear.
The "yellow band" and "red band" flavors offered direct drive plus a lower gear. The difference between the yellow and red models is in the brake. The red band had a disc-type brake, like the old New Departure coaster brakes, while the yellow band used a pair of brass shoes pressing against the inside of the shell, as with newer Bendix and other "modern" coaster brakes.
My friend had one on his stingray when we were kids by slightly back pedeling it would shift gears from low to high and back it worked pretty cool. Was a thing all kids wanted but few could afford. Those were the days.