I worry that your words and comments suggest you are conflating the system of study, the model of the system of study, and abstractions of the model. The particular ambiguities you describe come from mixing among these categories. Let us parse your wax burning example.
System, model, abstraction, interpretation, and semantics
The System: We have a candle made of wax. It burns. At various times, we measure the cumulative wax burnt. (Perhaps we actually measure some other physical property and infer the cumulative wax burnt from this measurement. This is an experimental detail that does not concern us further.)
The Model: Let $w$ be the quantity of cumulative wax burnt, $t$ be the time, $t_0$ be the time the burning started, and $t_1$ be the time the burning stopped. From the nature of burning in the system, $w$ is a continuous function of $t$. (This is not a mathematical claim. It is syntactically equivalent to "The quantity of cumulative wax burnt is a continuous function of the time", a statement about the physics of burning.) On theoretical grounds, $w$ is constantly zero before $t_0$, $w$ increases at a constant rate with respect to $t$ between times $t_0$ and $t_1$, and $w$ is constant for all times $t_1$ and later. During the time that $w$ increases at a constant rate with respect to $t$, we use the positive real parameter $a$ to denote the constant rate.
(A critical property of the model is that it attaches symbols to the quantities of interest in the system. Without this, the symbols and inferences appearing in the upcoming abstraction can never be related to the system. Additionally, any symbol used other than $w$, $t$, $t_0$, $t_1$, and $a$ cannot be attached to the system unless it is defined in terms of those symbols.)
(Notice that the model asserts "$w(t)$" will be physically meaningful, since the model asserts that the physical system is a process that converts time into cumulative burnt wax. "$t(w)$" will not be physically menaingful, since the physical system is not modeled as a process that converts cumulative burnt wax into time.)
The Abstraction: Let $T \subset \Bbb{R}$ be the minimal closed real interval containing the values of $t$ in the model and $W \subset \Bbb{R}$ be the minimal closed real interval containing the values of $w$ in the model. We have $w:T \rightarrow W$ defined by
$$ w(t) = \begin{cases} 0 ,& t \leq t_0 \\
a t ,& t_0 < t < t_1 \\ a t_1 ,& t_1 \leq t \end{cases} $$ with real-valued parameter $a > 0$ .
(There are no quantities in the abstraction. There is no time, no wax burnt, nothing about the experiment here. In fact, the abstraction is only attached to the experiment through the model so that the abstraction does not express anything about the system except what can be expressed through the symbolism established in the model.)
Alright, having performed that exercise, how can we find answers to your questions?
The experiment establishes that we will have a relation between the cumulative wax burnt and time. The construction of the experiment is such that for each time of measurement, there will be a single quantity of cumulative wax burnt. Since each time has a single quantity of cumulative wax burnt, we model cumulative wax burnt as a function (contra. relation) of time. In the abstraction, $w$ is a map from the real values which can be times to the real values that can be quantities of cumulative wax burnt. This is the sequence of steps that we use to express "cumulative wax burnt as a function of time", "$w$ as a function of $t$", and then $w:T \rightarrow W$. This sequence of steps means that we have an interpretation of expressions "$w(X)$" in the system, as long as $X$ is an element of $T$. If $X \not\in T$, "$w(X)$" is undefined in the abstraction and has no interpretation in the system.
In the abstraction, we can certainly differentiate $w(t)$ with respect to $t$ and obtain a piecewise function, $\frac{\mathrm{d}}{\mathrm{d}t} w(t) : T \smallsetminus \{t_0, t_1\} \rightarrow \{0,a\}$. But this is not the only thing we can do. In the abstraction, we can differentiate $w(t^2)$ with respect to $t$ and get
$$ \frac{\mathrm{d}}{\mathrm{d}t} w(t^2) = \left. \frac{\mathrm{d}}{\mathrm{d}s} w(s) \right|_{s = t^2} \cdot 2t \text{.} $$
In the abstraction, this is only valid for $t \in T$ where $t^2 \in T$. In the model, this is invalid : $t^2$ is not a time, it is a squared time; the model $w$ is a function of time, not squared time. So this calculation does not have an interpretation in the system.
So the short version is: in the abstraction, we are free to perform any valid mathematical manipulation we like. Such manipulations either satisfy the semantics established by the model and have an interpretation in the system or do not satisfy the semantics so do not have an interpretation. We can, in fact, write many things at the level of the abstraction, but to have an interpretation in the system, such writings must conform to the model.
Interpreting a function by altering its inputs
There is a particular abuse of this notion in Physics that may be enlightening. I'll establish up front that this example is exactly the opposite of what mathematicians prefer, and I think much of your question lies in the range between these two positions.
Say I have modeled a physical system as a function $f$ of position on a plane. For whatever reason, it is convenient to model position on a plane using Cartesian coordinates, with $x$ as the horizontal coordinate and $y$ as the vertical coordinate, and also using polar coordinates, with $r$ as the radial coordinate and $\theta$ as the azimuthal coordinate.
Note that the language of the model assigns the same interpretation to $f(x,y)$ and $f(y,x)$ because $f$ is a function of position and we have established that a pair of $x$ and $y$ (defined to have distinguishable semantics) is a position. If the model associates the same position to one $x$ and $y$ pair as it does to one $r$ and $\theta$ pair, then the model also established the same interpretation in the system to all four of $f(x,y)$, $f(y,x)$, $f(r,\theta)$, and $f(\theta, r)$. These equivalences are in the model, not the abstraction. But notice that this supplies an unambiguous interpretation to the question "What is the derivative of $f(x,y)$ with respect to $\theta$?" which interpretation very likely necessitates the answer is not zero.
When we pass from the model to the abstraction, we will fix the order of the arguments to $f$ so that $f(x,y)$ has an interpretation and $f(y,x)$ does not. Likewise we interpret $f(r,\theta)$ and not $f(\theta,r)$. (But, it is worth noting, we are free to abstractualize the order of arguments in whichever way is more convenient.) Now to the difference between physics and math.
A physicist looks at the two abstraction expressions $f(x,y)$ and $f(r,\theta)$ and sees the same $f$ as a function of position. A mathematician looks at the two abstraction expressions $f(x,y)$ and $f(r,\theta)$ and sees "the same procedure applied to the ordered pairs $(x,y)$ and $(r,\theta)$". These are very different interpretations of the same abstraction expressions. As a result, the answer to the question "What is the derivative of $f(x,y)$ with respect to $\theta$?" differs. For a physicist, one is asking how $f$ varies as its input is varied azimuthally near the Cartesian point $(x,y)$. For the mathematician, the answer is zero until we augment the model with a relation $(x,y) \leftrightarrow (r,\theta)$. (Those parenthetical lists are model positions, not abstraction ordered pairs.) Once that augmentation is in place, the mathematician interprets the question as "What is the derivative of $f(x(r,\theta),y(r,\theta))$ with respect to $\theta$?", implicitly using the model position to position relation to write Cartesian coordinates as a function of polar coordinates. The mathematician is likely to go one step further and write something like
$$ \tilde{f}(r,\theta) = f(x(r,\theta),y(r,\theta)) $$
to establish in the abstraction an explicit symbolic difference between the model $f$ that is a function of Cartesian coordinates and the model $f$ that is a function of polar coordinates. Then the question is translated to "What is the derivative of $\tilde{f}(r,\theta)$ with respect to $\theta$ expressed in terms of $x$ and $y$?"
I've actually been a little harsh in the above. Both viewpoints can be unified if we do not rush to coordinates. We could represent positions as vectors in a 2-dimensional real vector space in the abstraction, denoted $\vec{v}$. Then the only expression to consider is $f(\vec{v})$. Augmenting the abstraction by defining at each $\vec{v}$ a collection of four tangent vectors in the positive horizontal, positive vertical, positive radial, and positive azimuthal directions, all the apparent ambiguity in the above vanishes. This more accurately models the system, with $f$ as a function of position, not
as a function of ordered coordinates relative to some basis that is not dictated by the system. (Clearly. Because the model has two sets of coordinate systems.)
Summary
In attaching an abstraction to a system, we assign semantics to particular abstract expressions via a model. We are free to write any abstract expression we want, but such expressions need not have an interpretation relative to the semantics established by the model. The system relation "quantity one is measured with respect to quantity two" can be modeled as "$c$ represents quantity one, $d$ represents quantity two, and $c$ is a function of $d$". That model relation is then translated to the abstraction "$D$ is a set containing values of $d$, $C$ is a set containing values of $c$, and we have the function $f:D \rightarrow C:d \mapsto \dots$". This $f$ has the semantics endowed by the model of being a function of quantity two. We may abstractly treat this $f$ as a function of any abstract symbol. However, we risk losing an interpretation relative to the system if we do not write $f$ as a function of an expression have the interpretation of quantity two. We are abstractly allowed to differentiate this $f$ with respect to any expression, but we risk losing an interpretation relative to the system if we do not differentiate with respect to an expression having the interpretation of a quantity two.