Unified notion of what "$dx$" means
The symbol $dx$ in calculus is at first introduced just as a form of notation: in differentiation,
$$\frac{dy}{dx}$$
means the derivative of the function $y$ with respect to $x$. Letting $f(x) = y$, other notations are $f'(x)$, $\dot{y}$ (Newton's "fluxion" notation, used in physics), and $D_x[f]$ ("differential operator" notation).
It also appears in integration:
$$\int f(x)\ dx$$
means the anti-derivative of $f$ with respect to the variable $x$, and if the integral is a definite integral, again, it just denotes the variable with which we are integrating "with respect to".
So it seems that $dx$ is just notation, nothing more, nothing less. Yet then you may come across something weird like this:
$$\frac{dy}{dx} = x$$ $$dy = x\ dx$$ $$\int dy = \int x\ dx$$ $$y = \frac{x^2}{2} + C$$
Yet the above bunch of manipulations is, from the "notation" point of view, nonsense: "$dy$" and "$dx$" are not quantities, just pieces of notation. Saying $dy = \mathrm{something}$ is as ridiculous as saying $\sqrt{} = \mathrm{something}$. Yet it works.
Now, when we include more advanced math as well as the basic math, it seems we can compile the following list of answers as to "what $dx$ really is" -- and they're not one thing!:
It's just notation. That's what we just did. But then the above manipulation is nonsense. Someone said on this site that this was only a "particularly unenlightening" way to look at it.
It's a limit. $dx$ is what happens to $\Delta x$ as it arbitrarily small. Yet this is not rigorous enough -- to make the above manipulation work, it seems we have to pass out of the limit back to $\Delta y$ and $\Delta x$ since the limit of each is $0$.
It's a differential form. This gives the first "rigorous" definition. But here, we pull in manifold theory and non-Euclidean spaces to do our work, which while it is at home in advanced math, is definitely not something you'd want to put in Calculus class, where we are dealing with the conceptually much simpler Euclidean spaces, and if you want to make rigorous $dx$-manipulations in Euclidean space, seems like serious overkill. This also runs into problems with integration -- what happens if we are integrating discontinuous functions? This brings us to...
It's a measure. This definition seems limited only to integration. In this case, $dx$ in the integral stands for a measure -- a function which assigns a "size" ("length", "area", "volume", etc.) to subsets of the real line or Euclidean space, in particular the Lebesgue measure, which generalizes our usual, intuitive notion of "length", "area", etc. . Again, this is heavy as we have to delve into the deep structure of the real line but of course it's an advanced concept as well. Nevertheless, it doesn't require non-Euclidean spaces, and can be applied to discontinuous functions, but of course only works with integration (note you can't differentiate a discontinuous function).
It's an infinitely small number. This was how it was, in fact, originally conceived by the fathers of calculus. This approach was formalized rigorously by Abraham Robinson's non-standard analysis. This approach seems to be the heaviest of them all, as it requires probing down into mathematical logic to really understand the formalism -- in particular "model theory" and "ultrapowers" to extend the real line with infinitesimal numbers in such a way as to permit analysis to be done.
Now, it seems that 3) and 4) can be thought of as "differential" and "integral" notions of what "$dx$" means. 1) and 2) are not so helpful.
My question is: is there a single, unifying notion which ties all these together -- a "one true meaning of $dx$"? Or is "$dx$" a bunch of very, very different concepts, and thus it is silly to point to one or the other and call it the "real" meaning?
My opinion is that the quest for a universal meaning of such symbols as $\mathrm{d}x$ or $\mathrm{d}t$ is overrated. Of course the use of this symbols is motivated by history, as you noticed. However
- Derivatives can (and probably should) be denoted by a prime, or by the operator $\mathrm{D}$;
- Integration (like $\int_\Omega f \, \mathrm{d}\mu$) can and is denoted also by $I(f,\mu)$;
- the integral of the 1-form $\omega=f(x)\, \mathrm{d}x$ over the path $\gamma$ is $\int_\gamma \omega$;
- each single "manipulation" of differentials is rigorously justified by theory.
What I mean is that we could use $\partial$ for the differentiation operator acting on forms, $\delta$ for infinitesimals, and so on. We tend to use d mostly because we always try to minimize the number of symbols. Consider, for instance, the confusion about different integration theories. We write $\int$ for any integral, although we know that Riemann and Lebesgue integrals are not the same.
In the same way, we know that the d of differential forms and the d of integration theory are the same only in particular circumstances, but we prefer to abuse notation for the sake of comfort.