Fix a field extension $k\subseteq K$ (assume that the fields have characterostic $0$) and consider the two following definitions:

Now let's restrict our attention to a closed subscheme $X\subseteq\mathbb P^n_K$ (i.e. embedded projective variety), I have basically two questions:

• Definition $3.24$ in other terms means that I can find some homogeneous polynomials with coefficients in $k$ whose common zeros set is isomorphic to $X$. I' don't understand what is the practical meaning of definition $3.25$. I'd like a description in terms of common zeros of polynomials (remember: only in the case of embedded projective varieties).
• In definition $3.25$ it is very strange the fact that one requires the equality $$W= V\times_{\operatorname{Spec}k} \operatorname{Spec} K$$ but what happens if one requires the equality up to isomorphism? Namely: $$W\cong V\times_{\operatorname{Spec}k} \operatorname{Spec} K$$