You are trying to translate an english expression directly. I think 'a kind of' is an expression which is a deviation in english grammar. Glass is a kind of material ==> Glass is a material, (kind of). is more correct in grammar, though not used this way. You use korean same way. Glass is a material, (kind of) ==> 유리는 (일종의) 물질입니다. A mosquito is a bug (kind of). ==> 모기는 (일종의) 벌레입니다. A mosquito is a kind (of a bug) ==> 모기는 (벌레의) 한 종류 입니다. So, A is B means A는 B입니다. A is B (of C) means A는 (C의) B입니다. A is a student (of C school) ==> A는 (C 학교의) 학생입니다. Following example is same. But in english, you use a irregular expression. A new kind of pizza ==> 피자의 새 종류 (direct translation) But they are talking about pizza, not a new kind... (The subject is pizza) so, it had to be like this. A pizza, (of a new kind) ==> (새로운 종류의) 피자 (more natural korean) Well, I hope my explanation is correct. Let me know if anyone has different opinion.

의 is being uses as the with what you posted and of from the examples you have. meaning it is used multiple ways. (유리는 종류의 물질이에요 - It's the kind of glass material) (새로운 종류의 피자 - A new kind of pizza) (종류의 - Kind of) 의 - the). I hope this helps.