I think we are saying the same thing in different words, and I might be confusing "an alternative" with "a comparison". John compares R^2 with E^2, but RMSE can be considered an alternative to using R^2 in certain cases.
If you go back to the first line:
> People sometimes use R^2 as their preferred measure of model fit.
I think the post is going over why R^2 is not recommended as 2 is not only a measure of the error, but it includes a comparison with a constant model. John defines E^2 as a comparison metric which measures how much worse the errors are than if you used the true model.
Going back to a metric for determining model fit, RMSE/MSE/MAD are all alternative measures of model fit and are useful depending on the dataset.
If you go back to the first line: > People sometimes use R^2 as their preferred measure of model fit.
I think the post is going over why R^2 is not recommended as 2 is not only a measure of the error, but it includes a comparison with a constant model. John defines E^2 as a comparison metric which measures how much worse the errors are than if you used the true model.
Going back to a metric for determining model fit, RMSE/MSE/MAD are all alternative measures of model fit and are useful depending on the dataset.