The definition of flux and the energy-momentum tensor?

As pointed out in the question: Doubt regarding stress-energy tensor definition the energy-momentum tensor (aka stress-energy tensor) can be defined such that $T^{\mu \nu}$ is the flux of $P^\mu$ thru a surface of constant $x^\nu$. Timaeus (in their answer) defined the flux $T^{\mu \nu}$ such that the amount of $P^\mu$ passing through a surface of constant $x^\nu$ is given by: $$\Delta P^\mu=T^{\mu \nu}\frac{c\Delta t \Delta x \Delta y \Delta z}{\Delta x^\nu}$$ In which case the component $T^{00}$ becomes the energy density divided by c. I have however seen $T^{\mu \nu}$ given with different factors of $c$ such that $T^{00}$ is the energy density or mass density. In these cases are they defining the stress-energy tensor differently or the way they define flux? And is their a standard (agreed on) way to define either of these quantities.

I think your confusion primarily comes due to the $c=1$ convention used in relativity and high energy physics. When people do this, they don't really differentiate between a mass and its mass-energy, for example. This gets abused in all sorts of ways, hence the differing conventions of $c$ within $T^{\mu\nu}$. It turns out these definitions only differ by a factor of $c$, so as long as you make sure units work out to what you want them to be in the end, the precise difference is not important. The important thing to remember, though, is that all the components of $T^{\mu\nu}$ have the same units.