I too thought my hair was coarse or medium-coarse at first. There was a thread on this recently (I forget what it's called), but many of us think our hair feels coarse when we first start out. For some, this is because the hair can feel a little coarse when it is very dry and needs moisture. And if your hair is getting weighed down by emollient rich products, I think that's another sign it probably isn't coarse.
I suggest you pull a few dry, productless hairs out to check. Take just a few from different areas (temple, nape, etc.) and really look at them. I thought my hair was coarse until I did this, and was like... wait a second, these are pretty fine. I can easily see them because I'm brunette, but they are thin width-wise. Make sure you are looking at a single strand and not a few next to each other.
Also, I know Kathymack is low porosity and her hair loves protein. She has said (and I believe her, because I looked into it a little too) that naturallycurly.com is the only place she has read about low porosity hair disliking protein.