I am an amateur astronomer and science writer. I run a blog advocating the planet status of Pluto and all dwarf planets at laurelsplutoblog.blogspot.com .
The IAU is the problem. In 2006, four percent of the IAU, most of whom were not planetary scientists but other types of astronomers, voted that dwarf planets are not planets at all, essentially misusing Stern's term. He intended the term dwarf planet to be just like the term dwarf star. Within days, hundreds of planetary scientists signed a petition rejecting the IAU definition. They continue to reject it to this day. The IAU should never have touched this issue, as science is not decreed by "authority."
Pluto and dwarf planets are NOT "planetoids," as "planetoid" is a synonym for asteroid and refers to object not large enough to be rounded by its own gravity. The term "dwarf planet" was first coined by New Horizons principal investigator Alan Stern in 1991 to designate a third class of planets in addition to terrestrials and jovians. The question of whether dwarf planets are a subclass of planets remains a subject of ongoing debate.
The object you refer to as "Planet Nine," which is hypothetical, as it has not yet been discovered, should be called "Planet X," as the latter is the standard term for a hypothesized but undiscovered planet. It is not the solar system's ninth planet, as the solar system already has more than eight planets. Dwarf planets are planets too; they are simply a subclass of planets.