There has been much discussion over the last 100 years about registers. I myself was super confused for quite a long time. Well, there can be only two registers!
A register is series of homogeneous sounds produced by one muscle system that differ in quality from another series of homogeneous sounds produced by a different muscle system.
There is a chest register (arytenoid muscle system) and a falsetto register (crico-thyroid muscle system). Technically, there cannot be three registers because there are not three muscle systems that act upon the vocal cords, only two.
Now, as these two registers develop and coordinate, a singer has different sensations. There is not just chest voice or falsetto voice, but a third basic texture that emerges that can be called “full voice.” This third texture has been mistaken as a third register, but it is not. Rather, it’s appearance is from the healthy coordination of our two real registers, chest and falsetto.
In fact, our two registers can coordinate in a myriad of amazing ways to create a full palate of vocal colors and textures, all of which are useful for vocal expression.