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

![enter image description here][1]
![enter image description here][2]

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 $X$. Now 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:
$$W\cong V\times_{\operatorname{Spec}k} \operatorname{Spec} K$$
(It seems that in our case the two definitions coincide.)

 

  [1]: https://i.sstatic.net/LoGbd.jpg
  [2]: https://i.sstatic.net/VqUFE.jpg