If you have a point "A" in space, and another point "B" in space, 10 light years away in relatively flat space with regard to gravitational waves. Then you place a neutron star directly in the middle of point A and B. Now the distance between point A and point B is greater than 10 light years because of time dilation. Is that right?
Your question If mass density curves space-time, then why isn't density (at each $x$, $y$, $z$) considered a dimension in space-time? about a more dimension for gravitational potential is a reasonable question I want to try to explain on an example what perhaps do you mean.
Take a thin and flat metal sheet and heat it up at one edge. The sheet will not be more flat, it will be humpbacked. The density of the heated material is lower than of the cold(ear) material and the expansion of the hotter material is humpbacked.
What @dmckee suggested in his answer to your question befo this one is to add coefficients to the dimension and by this way to describe the space. For my example this means that the coefficients are functionally connected with he heat distribution on the sheet. But the sulu tigon is much more complicated. Even for a round sheet and a hot spot in the centre you have to include at minimum the really existing stress of the sheet. (And only if you perform special stress before heating (say by hammering radial pattern into the sheet) you can be sure - more or less - that the sheet under heating will be humpbacked how you calculated it before.)
Einstein has suggested the same for gravitational potential. In his model the speed of light is a constant value. What often is missed that this means that we will measure the same value for any positions in space with different gravitational potential, but a third observer in space with a different potential will admit a difference in how fast clocks are running. Than higher the potential than slower the clock runs. For your example the light takes longer if there is a massive body near your path between the masses A and B for two reasons: the light gets deflected by the third body and is not more straight and the lights speed - for an "observer in the high" this is measurable - is slower than it was before the third body was placed.
But as @dmckee commented
Asking about the spatial distance along the straight line is possible but requires you to chose a frame of reference in which to ask it as there is no unique answer.