Is a generated field independent of the extension over which it is generated?

This sounds like a very basic question of field theory. It’s probably a small detail, most authors and introductory sources to field theory seem to ignore it.

Given a field extension $ L/K $ and a subset $S$ of $L$, there is a smallest subfield of $L$ that contains $K$ and $S$. It is the intersection of all subfields of $L$ that contain $K$ and $S$, and is denoted by $K(S)$. My confusion is about this notation $K(S)$ which does not denote explictly $L$. The first time I read this definition, I wanted to write it $K_L(S)$ because I don’t know whether it depends on $L$. But most sources just write $K(S)$ and that’s where my confusion is, because they seem to implicitly assume that it somehow does not depend on $L$ which I don’t find obvious.

My question is thus: If I have two extensions $ L/K $ and $ L’/K $ over the same field $K$, and $Ssubseteq Lcap L’$, is it true that $K_L(S) = K_{L’}(S)$ or at least that they are field isomorphic ?

I am not assuming any relationship between $L$ and $L’$, I am assuming the minimum for $K_L(S)$ and $K_{L’}(S)$ to be defined, $Lcap L’$ might not be a field at all.

If the two extensions form a tower, for example $Lsubseteq L’$ then I think it is true because $K_L(S)$ represents the smallest set closed under a finite number of field operations. However if $L$ and $L’$ are not a tower, i do not have an answer. I don’t even know if the sum/product in $L$ would correspond to the sum/product in $L’$ for two elements coming from $S$.
For that reason, i find it confusing to use the notation $K(S)$ without any explanation, which most sources do.

It seems like an important thing to understand, because I have noticed that most people then usually, don’t think of the extension $L$ at all, they seem to just think of $K(V)$ as "adjoining" some elements $V$ into a field $K$, and they really seem to think that $K(V)$ as if the concept could be defined just from $K$ and $V$. However my attempt to define "an extension generated from a set $V$ and a field $K$" just fails, without $L$ because i need some underlying operations to be defined.

EDIT: Thanks to Lee Mosher for his answer: Yes $K_L(S)$ depends on $L$.

I think a motivation for writing only $K(S)$ is the following:

So far what I have defined is, a generated subextension $K_L(S)$.
We can actually extend the concept to a generated extension.
An extension $ L/K $ is said to be generated by $ Ssubseteq L$ if $ L=K_L(S) $.
According to that definition we can notice that $K_L(S)$, as an extension, is generated by $S$, because $K_L(S) = K_{K_L(S)}(S)$, which works because $K rightarrow K_L(S) rightarrow L$ is a tower of extensions.
Therefore it makes sense to just write $K(S)$ because it is constant in a given tower, and it avoids writing those packed subscripts.
Also that way, concepts like a simple extension can be elegantly defined as an extension generated by one of its elements and written $K(alpha)$ (but truly it still depends on L).