This is a tremendously common confusion to have, and in my experience, people are notoriously bad at explaining this concept. I'm sorry that you had to deal with people who were abrasive in addition to poor expositors.
In an arbitrary vector space, you cannot talk about components. They actually don't exist. Now, you can impose them on a finite-dimensional space by providing a bijective linear transformation from the arbitrary vector space to $F^n$, but then they're just that: an imposition, because any other bijective linear transformation will choose different would-be "components".
Components exist in $F^n$ because of the actual nature of the objects involved. So you don't need a basis, you can just look at an arbitrary object $(a,b,\dots,n)$, and find any of its components, because they're built into the object. This can be confusing because we also write coordinate vectors in this way, and when the basis is the standard basis, there is no difference between the components and the coordinates. However, in any other basis, there will be a difference.
(Edit: Val made an important point in the comments. I should have been more careful when I said there was "no difference". The fact is that coordinates and components are never conceptually the same, but I meant to say that in the standard basis case they will be numerically equal.)
Lacking a basis at all, you might want to say that $F^n$ still has coordinates implied by its components. But, in my opinion, this seems silly, since you cannot do the same in other spaces.
So the short answer is: Yes, there is a difference, because components are part of the objects.
As for your "collection of vectors" notion, they are basically the same. But it is easy to imagine a collection of vectors which is not a vector space: for example the circle in $\mathbb{R}^2$. This is definitely a collection, and the objects in it are definitely vectors, but it is not a vector space.
What I assume you meant by "collection" was what we might call a "meaningfully structured collection", and the meaningful structure is described precisely as an abelian group over which elements can be scaled by objects in a field. In that sense, your notion is correct, though a bit less transparent.
As a student of math and physics, this has been one of the biggest annoyances for me; I'll give my two cents on the matter. Throughout my entire answer, whenever I use the term "function", it will always mean in the usual math sense (a rule with a certain domain and codomain blablabla).
I generally find two ways in which people use the phrase "... is a function of ..." The first is as you say: "$f$ is a function of $x$" simply means that for the remainder of the discussion, we shall agree to denote the input of the function $f$ by the letter $x$. This is just a notational choice as you say, so there's no real math going on. We just make this choice of notation to in a sense "standardize everything". Of course, we usually allow for variants on the letter $x$. So, we may write things like $f(x), f(x_0), f(x_1), f(x'), f(\tilde{x}), f(\bar{x})$ etc. The way to interpret this is as usual: this is just the result obtained by evaluating the function $f$ on a specific element of its domain.
Also, you're right that the input label is completely arbitrary, so we can say $f(t), f(y), f(\ddot{\smile})$ whatever else we like. But again, often times it might just be convenient to use certain letters for certain purposes (this can allow for easier reading, and also reduce notational conflicts); and as much as possible it is a good idea to conform to the widely used notation, because at the end of the day, math is about communicating ideas, and one must find a balance between absolute precision and rigour and clarity/flow of thought.
btw as a side remark, I think I am a very very very nitpicky individual regarding issues like: $f$ vs $f(x)$ for a function, I'm also always careful to use my quantifiers properly etc. However, there have been a few textbooks I glossed over, which are also extremely picky and explicit and precise about everything; but while what they wrote was $100 \%$ correct, it was difficult to read (I had to pause often etc). This is as opposed to some other books/papers which leave certain issues implicit, but convey ideas more clearly. This is what I meant above regarding balance between precision and flow of thought.
Now, back to the issue at hand. In your third and fourth paragraphs, I think you have made a couple of true statements, but you're missing the point. (one of) the job(s) of any scientist is to quantitatively describe and explain observations made in real life. For example, you introduced the example of the amount of wax burnt, $w$. If all you wish to do is study properties of functions which map $\Bbb{R} \to \Bbb{R}$ (or subsets thereof), then there is clearly no point in calling $w$ the wax burnt or whatever.
But given that you have $w$ as the amount of wax burnt, the most naive model for describing how this changes is to assume that the flame which is burning the wax is kept constant and all other variables are kept constant etc. Then, clearly the amount of wax burnt will only depend on the time elapsed. From the moment you start your measurement/experiment process, at each time $t$, there will be a certain amount of wax burnt off, $w(t)$. In other words, we have a function $w: [0, \tau] \to \Bbb{R}$, where the physical interpretation is that for each $t \in [0, \tau]$, $w(t)$ is the amount of wax burnt off $t$ units of time after starting the process. Let's for the sake of definiteness say that $w(t) = t^3$ (with the above domain and codomain).
"Sure, $w$ only has the interpretation we think it does (cumulative amount of wax burnt) when we provide a (real number in the domain of definition, which we interpret as) time as its argument"
True.
"...Sure, we can't really interpret $w$ in the same way if I did this, but there is nothing in the definition of w which stops me from doing this."
Also true.
But here's where you're missing the point. If you didn't want to give a physical interpretation of what elements in the domain and target space of $w$ mean, why would you even talk about the example of burning wax? Why not just tell me the following:
Fix a number $\tau > 0$, and define $w: [0, \tau] \to \Bbb{R}$ by $w(t) = t^3$.
This is a perfectly self-contained mathematical statement. And now, I can tell you a bunch of properties of $w$. Such as:
- $w$ is an increasing function
- For all $t \in [0, \tau]$, $w'(t) = 3t^2$ (derivatives at end points of course are interpreted as one-sided limits)
- $w$ has exactly one root (of multiplicity $3$) on this interval of definition.
(and many more other properties). So, if you want to completely forget about the physical context, and just focus on the function and its properties, then of course you can do so. Sometimes, such an abstraction is very useful as it removes any "clutter".
However, I really don't think it is (always) a good idea to completely disconnect mathematical ideas from their physical origins/interpretations. And the reason that in the sciences people often assign such interpretations is because their purpose is to use the powerful tool of mathematics to quantitatively model an actual physical observation.
So, while you have made a few technically true statements in your third and fourth paragraphs, I believe you've missed the point of why people assign physical meaning to certain quantities.
For your fifth paragraph however, I agree with the sentiment you're describing, and questions like this have tortured me. You're right that $w$ is a function of a single variable (where in this physical context, we interpret the arguments as time). If you now ask me how does $w$ change in relation to the distance I have started to walk, then I completely agree that there is no relation whatsoever.
But what is really going on is a terrible, annoying, confusing abuse of notation, where we use the same letter $w$ to have two differnent meanings. Physicists love such abuse of notation, and this has confused me for so long (and it still does from time to time). Of course, the intuitive idea of why the amount of wax burnt should depend on distance is clear: the further I walk, the more time has passed, and hence the more max has burnt. So, this is really a two step process.
To formalize this, we need to introduce a second function $\gamma$ (between certain subsets of $\Bbb{R}$), where the interpretation is that $\gamma(x)$ is the time taken to walk a distance $x$. Then when we (by abuse of language) say $w$ is a function of distance, what we really mean is that
The composite function $w \circ \gamma$ has the physical interpretation that for each $x \in \text{domain}(\gamma)$, $(w \circ \gamma)(x)$ is the amount of wax burnt when I walk a distance $x$.
Very often, this composition is not made explicit. In the Leibniz chain rule notation
\begin{align}
\dfrac{dw}{dx} &= \dfrac{dw}{dt} \dfrac{dt}{dx}
\end{align}
Where on the LHS $w$ is miraculously a function of distance, even though on the LHS (and initially) $w$ was a function of time, what is really going on is that the $w$ on the LHS is a complete abuse of notation. And of course, the precise way of writing it is $(w \circ \gamma)'(x) = w'(\gamma(x)) \cdot \gamma'(x)$.
In general, whenever you initially have a function $f$ "as a function of $x$" and then suddenly it becomes a "function of $t$", what is really meant is that we are given two functions $f$ and $\gamma$; and when we say "consider $f$ as a function of $x$", we really mean to just consider the function $f$, but when we say "consider $f$ as a function of time", we really mean to consider the (completely different) function $f \circ \gamma$.
Summary: if the arugments of a function suddenly change interpretations (eg from time to distance or really anything else) then you immediately know that the author is being sloppy/lazy in explicitly mentioning that there is a hidden composition.
Best Answer
With regard to a function in the context given, the phrase spherically symmetric means that the function, which is a function of a vector, depends only on the magnitude of that vector. That is, $$ f(x) = f(y) \qquad\text{whenever}\qquad \|x\| = \|y\|. $$ There are other equivalent ways of describing this notion, which can be stated at various levels of rigor (in each statement, it might help to imagine that $n=3$, as that fits the context of the question):
Given the context provided in the question, it may be worth mentioning that the wave equation in three dimensions is solved by a function which is spherically symmetric in the sense given above.
[1] I think that, in general, "rotation" is equivalent to "$SO(n)$ action", i.e. a rotation is the action of an element of the special orthogonal group, which is basically the set of $n\times n$ dimensional matrix with determinant $1$. The orthogonal group consists of matrices with determinant $\pm 1$. These matrices give rotations (if the determinant is $1$), plus rotations with reflection (if the determinant is $-1$). A spherically symmetric function is invariant under reflection so, hence the description above.