This is just my personal opinion, as there is no designated 一覧 of the types of fairies.
There are 5 main types of fairies: light, night, water, nature, and air. Each of them has sub-categories, which are as follows:
Light: Fire, Sun, Common Light
Night: Moon, Star, Midnight
Water: Ocean, Snow and Ice, Freshwater, Rain, Mist
Nature: Forest, Flower, Animal
Air: Wind, Storm, Common Air
The 妖精 of the different types have a bit of control over whatever their type is. For example, a rain fairy would have control over the rain. Not complete control. They don't make the rain. They could do simple things like decide how fast it falls, where it goes, and how it splashes on the ground. Their clothing also reflects their type, and so does their personality.