I'm trying to have a conversation so I can figure it out. Pretty confident that being a classifier does not make it machine learning, econometrics has classifiers too. Econometric models also need data to train them, so I'm not sure your second point is helpful either. Unless you're claiming the difference is nothing but whether the model is used by an economist.
More generally, I recommend Breiman's two cultures article for some insight into the similarities and differences.
If you need a really simple explanation, then machine learning is a tool for generating predictions, while statistics is a method for performing inference about causes.
