Mean Squared Error incompatible definitions

Question

Let $x$ be a vectorial estimator, and we aim to estimate $y$. Then the mean squared error can be defined as $MSE(x) = \mathbb{E}[\|x-y\|^2]$. On the other hand, in image processing the mean squared error for deterministic vectors $x,y$ is defined as $MSE(x,y) = \frac{1}{n}\|x-y\|^2$, where $n$ is the dimension of the vectors. My question is - shouldn't there also be a normalization of $\frac{1}{n}$ in the probabilistic case? It would make sense if the two are to be consistent, but I haven't seen a definition that normalizes by $1/n$ in the probabilistic case.