How to interpret morphisms of field extensions? In field theory the following definition is (as far as I can tell) standard:

Let $K$ be a field. Given two extensions $K\subset L$ and $K\subset L'$, we say that they are $K$-homomorphic if there exists a field homomorphism $\varphi:L\to L'$ such that $\varphi$ is the identity on $K$.

I have some trouble with this definition. Namely, it makes sense to me when $K,L,L'$ all happen to  simultaneously be subfields of some "universal field" $U$ where everything is taking place (e.g. extensions of $\mathbb{Q}$ in $\mathbb{C}$), but when it's about "abstract" extensions, without reference to a common enveloping field, the technicalities start to pester me (e.g. how is it possible for the same set $K$ to be a subset of two other unrelated sets? where is it all "taking place"?). I tried then to re-interpret this definition in greater explicitness, considering instead of literal inclusion $K\subset L$ homomorphisms $i:K\to L$ and $i':K\to L'$ (which are automatically injective) and calling $\varphi$ an extension morphism if $i'\circ\operatorname{id}_K=\varphi\circ i$ (the diagram commutes). This then leads to its own complications, because now that $K$ is not a literal subfield of $L$, stuff like $K(S)$ needs to be reinterpreted and some basic lemmas feel more complicated as a result. But then if I switch back and consider something "concrete" like $\mathbb{Q}\subset\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}\subset\mathbb{Q}[x]/(x^2-2)$, somehow it's not unnatural to think of $\mathbb{Q}$ as existing in two different places simultaneously. I know that for algebraic extensions of $K$ you can think of it all as taking place in an algebraically closed field $C$ containing $K$ (e.g. how it is with $\mathbb{Q}$ and $\mathbb{C}$), but then I start wondering how that all depends on the way $K$ might be embedded in $C$ and how many embeddings might there be and so on.
That's not to say that fields have me completely confused - when the lines of inclusion are clearly delineated, I don't have any problems. For example, everything related to things happening  in the context of a single pair $K\subset L$ (algebraicity, separability, degree, etc.) is clear to me. I also have no trouble with such pairs interacting, e.g. where you consider things like $K\subset L,K'\subset L'$, an isomorphism $\varphi:K\to K'$ and its various extensions to $L$. It's when the lines start to get blurry (e.g. in the previous setting, when $\varphi$ is taken to be $\operatorname{id}_K$) that I start to get confused.
Anyway, sorry about the ramble. The long and short of it is, field extensions and their morphisms are a bit of a salad in my head and I was hoping to get some explanations on how to interpret it all.
 A: 
but when it's about "abstract" extensions, without reference to a common enveloping field, the technicalities start to pester me (e.g. how is it possible for the same set $K$ to be a subset of two other unrelated sets? where is it all "taking place"?).

You can write everything down in terms of homomorphisms (which are automatically injective) $i : K \to L$ and $i' : K \to L'$ as you suggest. Then a $K$-homomorphism is a homomorphism $\varphi : L \to L'$ such that $\varphi \circ i = i'$ as you say (there is no need to explicitly compose with the identity morphism). I don't know what you mean by "stuff like $K(S)$ needs to be reinterpreted" but everything continues to work just fine.
There is no issue with thinking of $K$ as literally a subfield of every field extension of $K$; the point of the definition of a $K$-homomorphism is exactly to guarantee that you don't run into any trouble by doing this, because the definition forces all computations done in all such copies of $K$ to be consistent (as long as you only ever move from one field to another via a $K$-homomorphism).

But then if I switch back and consider something "concrete" like $\mathbb{Q}\subset\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}\subset\mathbb{Q}[x]/(x^2-2)$, somehow it's not unnatural to think of $\mathbb{Q}$ as existing in two different places simultaneously.

For $\mathbb{Q}$ the condition is automatic; that is, if $L, L'$ are two field extensions of $\mathbb{Q}$ then every homomorphism $L \to L'$ is automatically a $\mathbb{Q}$-homomorphism. This is a good exercise.
The condition wouldn't be automatic already for $K = \mathbb{Q}(i)$, say, and would then need to be checked.

I know that for algebraic extensions of $K$ you can think of it all as taking place in an algebraically closed field $C$ containing $K$ (e.g. how it is with $\mathbb{Q}$ and $\mathbb{C}$), but then I start wondering how that all depends on the way $K$ might be embedded in $C$ and how many embeddings might there be and so on.

It's good that you're worrying about this and the answer is to not try to embed everything into a common field, it's not necessary and ultimately makes things more confusing because you're making extra choices you don't need to.
