we can define a field extension as $K$ is a subset(!) of $L$ both fields, and the same $1$ and same operation (just restricting). But The Stacks project defines a field extension as a morphism $i: K\to L$, which is necessarily injective. Hence one can identify $K$ with $i(K)$ as a subfield of $L$.
Is this more general definition compatible with all results of Galois Theory? For example one can define separability for $i: K\to L$ in the obvious way, which will be equivalent to $L/i(K)$ is separable. Or should I just ignore this definition because I have always to carry the injective morphism with me (imagine we have a tower of extensions... That would be annoying.)