In general as you noted it is equivalent to either make opacityInput = yGradient, or make opacityInput = density, density = yGradient, and densityScale 1.0. However the density could also be used for things like colorInput. At any rate there are a lot of ways to hook things up and some will have the same effect.
The input settings on the shading attributes(color, opacity, incandescence) will never affect the dynamic simulation. However doing something like making temperature a gradient could affect the simulation. As well doing this with density could also affect the simulation… density buoyancy would add the buoyancy into the velocity grid each step based on the gradient. Even though there would then be no density grid, one could propagate other grids, like temperature or fuel and potentially base opacity on those grids instead of density.
If you are not doing a dynamic simulation, but just texturing a volume then you are probably best off using the shading input gradients directly instead of defining density as a gradient.
In terms of coordinate speed: The fluid solve propagates the various grid values through the fluid based on the fluid velocity. Note that the velocity could even be a gradient and not a grid. All the various grids… density, temperature, color, coordinates, fuel and even velocity… are propagated through the fluid. Velocity is a special case because it propagates its self, which can lead to instability if it is not carefully computed. The coordinate grid has the added control “coordinate speed” which basically scales the velocity when moving coordinates. This is useful because as a simulation develops the coordinates pull apart and shear which changes the look of the texture. Lets say your texture was a nice photo of a house made into a wet oil painting. Moving the coordinates is like smearing the paint around with your fingers. In terms of clouds, nice pluffy clouds will quickly become streaky swirls and loose their character. However a little motion to follow the velocity can help keep the clouds from looking fixed in space.
The texture time animation is very different and does not pick up any of the motion of the flow. Usually when using a coordinate grid I find it helpful to make the coordinate speed low and also slowly animate texture time. Also if the fluid flow has a dominant direction one might also animate the texture offset and rotate to more closely match. However if your flow is fast or the animation long then you might find the coordinate grid doesn’t work that well. (texturing is no substitute for a high resolution simulation)
However something else that helps when using coordinate grids is to keep the simulation relatively low resolution and viscous. This keeps the grid from stretching too much. It might also help to use multiple fluids and animate the positions of fluid containers. (it all depends on the scene)
