Throughout this answer, we're of course going to use the modern definition of a function as a "rule" which has a certain domain and target space.
You're not missing anything, and your interpretation of the function notation is perfectly correct. What is going on is that the other students and Professors are unfortunately abusing notation, where they're using the same symbol $F$ in two different places with different meanings. So, it's not your fault for being confused. It doesn't matter what letters we call things. If $F(t)=4t^2$, then it doesn't matter that you define $s(t)=2t$. We still have $F(s)=4s^2$. That's literally the definition of the notation of functions.
So, how do we formally express what the Prof/students are saying? Well, define the mapping $\phi:\Bbb{R}\to\Bbb{R}$ as $\phi(t)=2t$. This is an invertible function with the inverse given as $\phi^{-1}:\Bbb{R}\to\Bbb{R}$, $\phi^{-1}(s)=\frac{s}{2}$. Here, don't pay attention to the actual letters used. I could have said $\phi$ is the function such that for all $@\in\Bbb{R}$, $\phi(@)=2@$, and that $\phi^{-1}$ is the inverse function, which means for each $\sharp\in\Bbb{R}$, we have $\phi^{-1}(\sharp)=\frac{\sharp}{2}$. THe choice of letters has nothing to do with the definition of the functions. Of course we should use good notation, but from a logical perspective, there's nothing stopping us from using anything we want (you may want to read this answer of mine for more remarks about this).
So, if $F:\Bbb{R}\to\Bbb{R}$ is the function $F(t)=4t^2$, then $F\circ \phi^{-1}:\Bbb{R}\to\Bbb{R}$ is the function defined as for each $s\in\Bbb{R}$,
\begin{align}
(F\circ \phi^{-1})(s)&=F\left(\phi^{-1}(s)\right)=F\left(\frac{s}{2}\right)=4\left(\frac{s}{2}\right)^2=s^2.
\end{align}
(In your post you wrote $F(s)=2s$, when you probably meant $s^2$). Anyway, this is the correct statement. What this is saying is that if we express $F$ in terms of new variables (i.e by composing with $\phi^{-1}$), then we get the new function $F\circ \phi^{-1}$, which is just the mapping $s\mapsto s^2$.
As a general remark, if in some physics discussion, if you're talking about a certain function, and then suddenly we're saying it's a function of some other quantitity, then it means the author has omitted an extra composition in the notation. For example, if you're talking about motion of a particle in one-dimension and talk about "force as a function of time" (in the above example this would be the function $F$), and then in the middle of the discussion, we talk about "force as a function of position" (in the above example this would be the function $F\circ \phi^{-1}$), then it means the author is considering a new function by appropriately composing with $\phi$ or $\phi^{-1}$, where $\phi$ is the mapping from time to position. In your post, you used the notation $s(t)=\cdots$. I intentionally avoided this and called the function $\phi$ because I don't like to mix up the names of the functions with the arbitrary letters I use for elements in the domain/target (such as $t$ or $s$).
Also, it is common in physics/differential geometry to consider one and the same function but "expressed in different coordinates". For example, you might hear about "$f$ as a function of coordinates $x_1,\dots, x_n$" vs "$f$ as a function of coordinates $y_1,\dots, y_n$". Perhaps this might be too abstract for now, but bear with me. The meaning of these statements is that we start with some arbitrary set $M$, and suppose you have a given function $f:M\to\Bbb{R}$. Now, suppose also that you're given two invertible mappings $\alpha:M\to\alpha[M]\subset\Bbb{R}^n$ and $\beta:M\to\beta[M]\subset\Bbb{R}^n$.
Now, we have three mappings in the game $f$, $\alpha$ and $\beta$. We can now construct two new functions $f_{\alpha}:=f\circ \alpha^{-1}:\alpha[M]\subset\Bbb{R}^n\to\Bbb{R}$ and $f_{\beta}:=f\circ \beta^{-1}:\beta[M]\subset\Bbb{R}^n\to\Bbb{R}$. We think of $f_{\alpha}$ as "$f$ expressed in terms of $\alpha$ coordinates" and $f_{\beta}$ as "$f$ expressed in terms of $\beta$ coordinates". There is obviously a relationship between $f_{\alpha}$ and $f_{\beta}$, namely
\begin{align}
f_{\beta}&=f_{\alpha}\circ (\alpha\circ \beta^{-1})\tag{$*$}
\end{align}
This is a proper equality of functions, and all the compositions are well-defined etc.
In physics you might hear this along the lines of "if we know $f$ as a function of $y=(y_1,\dots, y_n)$ and we know $y$ as a function of $x= (x_1,\dots, x_n)$ then we know $f$ as a function of $x$", something like "if $f=f(y)$ and $y=y(x)$ then $f=f(y)=f(y(x))=f(x)$". I find this sort of language and notation ambiguous because we're using the same symbol $f$ with several different meanings and also an equality like $f=f(y)$ is actually meaningless (I know it it meant to emphasize that $f$ depends on $y$ but it's just formally wrong).
But hopefully it's clear now that equation $(*)$ is what is meant, and all we're doing is composing the actual function $f$ appropriately.
