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What is a good mental representation of $$dt$$ vs $$d\tau$$?

I'm not quite sure I understand the question. Here's how I "mentally" interpret $$t$$.

$$t$$ is a coordinate. I wouldn't actually picture it as "time flowing", as I wouldn't picture the coordinate $$r$$ as "distance from the center". I would picture it simply as a coordinate, used to describe your world. So, in a sense, we may say it's common to all the observers. For example, instead of $$(t,r,\theta,\phi)$$ you may decide to use the Kruskal–Szekeres coordinates $$(T,X, \theta, \phi)$$, where here the time coordinate is $$T=T(t,r)$$. So, should we use $$t$$ or $$T$$, what is our "real" time, which is the best one? None of them. Depending on what physical situation you are analyzing, a set of coordinates may be more useful than another one, describing the physical context in a clearer way. Coordinates are a tool you are using to describe your world.

Let's focus on the Schwartzchild metric, as it's the most simple one: $$$$ds^2=\Big(1-\frac{2m}{r}\Big)dt^2-\Big(1-\frac{2m}{r}\Big)^{-1}dr^2-r^2d\Omega^2$$$$ which is asymptotically flat (i.e. we recover the Minkowski metric far away from the origin). In this case, $$dt$$ has a physical meaning asymptotically, as it represents the proper time measured by an observer at rest at $$r\to\infty$$. So we may say that $$t$$ represents a "real time" only for an observer at rest at infinity.

Let's make an example to clarify the difference between $$dt$$ and $$d\tau$$ and how they are related. Let's consider a photon in a circular orbit ($$r=3m$$) in the plane $$\theta=\pi/2$$. This is a light-like event, so we have $$ds^2=0$$, thus computing from the Schwartzschild metric, we get $$$$0=\frac{1}{3}dt^2-9m^2d\phi^2\implies dt^2=27m^2d\phi^2$$$$ Now, say we want to know how much time passes for an observer who is at rest at $$r=3m$$. You can picture it this way: an observer at $$r=3m$$ "shoots" a photon with a laser and he starts a chronometer, and he stops it as he sees "his" photon again, once the photon has successfully completed the orbit. What we need is the relation $$$$d\tau_{r=3m}=\frac{1}{3}dt^2$$$$ obtained by setting $$dr=d\phi=d\theta$$ (as the observer is at rest) and $$r=3m$$. Substituting $$dt^2=27m^2d\phi^2$$ in the equation and integrating from $$0$$ to $$2\pi$$ we obtain $$\Delta\tau_{r=3m}=6\pi m$$. What would a distant observer at rest say about an event "taking" $$dt^2=27m^2d\phi^2$$ coordinate time? Again, from the Schwartzschild metric $$$$d\tau_{r=\infty}=dt^2\implies \Delta\tau_{r=\infty}=6\pi m\sqrt{3}$$$$

I hope I got the point of your question, at least partially.