Does it matter if the scaling term is in the DFT versus the inverse DFT? From Wikipedia, the equation for the 1D DFT is

From a separate source, the equation for the 2D DFT is

Notice how the 2D DFT definition features a scaling term while the first definition does not. Is this just an artifact of where the scaling takes place between the DFT and inverse DFT? Does it matter which transform contains this scaling term?
It's mostly a matter of convention.
There are three options. $1/N$ for the inverse transform, $1/N$ for the forward transform and $1/\sqrt{N}$ for both. The first is the one most commonly used, the last one has the advantage that Parseval's theorem holds directly, i.e. $$\sum |x_n|^2 = \sum |X_k|^2$$