This is where we have a problem with the meaning of words. Sometimes we mean one thing in certain circumstances, and yet another when we are talking about something else. We often interchange them, and they really have quite specific meanings.
In relation to hair, we speak of it being thick or thin, and coarse or fine.
When we are talking about density, we refer to the number of hairs on the head, usually expressed as number per square inch. If you have a very large number of hairs, then the hair is called "thick". If you have few hairs, and can see the scalp easily, the hair is "thin".
When we talk about texture, we are referring to the individual size and the feel of the hair, whether it's extremely skinny or very fat. So if the hair is very small in size, we say the hair is "fine". If each hair is quite fat, we say the hair is "coarse". Think of coarse and fine sand, or sugar.
So it is possible for someone to have very few hairs on the head, but each hair is fat, making it thin and coarse. Someone can have a lot of hairs, but each one is very skinny, making it thick and fine (that is the best to work with for styling purposes). They may both have the same size of ponytail, because of the volume they take up.
Most people mix these words up and that is where the confusion comes from. And it's really hard to describe a hair as being thin (skinny) because thin means few in number.
So your stylist is right: you have fine hair (each is skinny and silky) while it is also thick since you have a lot of it.