Two different definitions of Grassmannian

Question

there are two different notations of the Grassmannian and it confuses me. One definition is given by a representable functor from schemes to sets, see for example Stacks Project section 27.22. Another definition, over a field, says that $G(k,V)$ is the set of all $k$-dimensional subspaces of $V$. How these definitions are related? Is there any reference which connects between these definitions? Moreover, as a special case of Grassmannians, the projective space as a scheme consists not only maximal ideals generated by homogeneous linear polymials. It should consist of all homogeneous prime ideals of the polynomial ring. Hence these two definitions contradict each other. I wonder why ignoring other points of the projective spectrum makes sense. Yoav.