Choice free method to define immersions into Projective Space

Let $ X$ be a variety and $ \mathcal{L}$ be a very ample line bundle on $ X$ . Suppose $ H^0(X,\mathcal{L}) = \langle s_0,…,s_n \rangle$ then there is an immersion into projective space:

$ $ X \rightarrow \mathbb{P}(\,H^0(X,\mathcal{L})\check \,)$ $

given by evaluation on closed points $ x\rightarrow \{s\in H^0(X,\mathcal{L})\,|\,s(x)=0\}$ .

My first question is this: how can I extend the definition of this morphism to non-closed points? I know there is alternative approach as in Hartshorne, however I was looking for a choice free method to define such morphism.

My second question is somewhat related: it is often said that $ \mathcal{L}$ gives rise to a morphism

$ $ X \longrightarrow \mathrm{Proj}\left(\oplus_{k\in\mathbb{N}} \,H^0(X,\mathcal{L}^{\otimes k}\,)\right)\longrightarrow \mathbb{P}_k^n = \mathrm{Proj}\left(\,k\,[s_0,s_1,…,s_n]\,\right) $ $

where I believe the first map is an isomorphism (but I am not sure) and the second map is an immersion into projective space (again I am not sure you would want to define projective space in this way). How can you see this fact? I cannot find any references so feel welcome to just tell me where to look it up.

Thank you!