Overfitting can happen in many ways -- your training objective can be different at train and test time, or as you suggest the datapoints you use can be different at train and test time.
For overfitting induced by datapoints: If you include the datapoints in your problem specification, then you can say they induce bias at test time. If you treat the choice of training datapoints as a random variable, separate from the problem specification, then you can say they induce variance at test time. The difference is essentially semantic though. In general, you can freely move contributions to the error between bias and variance terms by changing which aspects of the modeling framework you define as fixed by the problem definition, and which you take to be stochastic.
And the reply also leaves me with a similar impression:
> your training objective can be different at train and test time
But this is not overfitting, this is concept drift, a very different and well-defined thing in ML.
> the datapoints you use can be different at train and test time
Both train and test data came from the same population. They are just different incomplete random samples.
I guess what I am getting at - overfitting happens because we know we are training a model on an incomplete representation of the whole. But that representation is not a proxy, as suggested in the article - it is not slightly different to the goal. It's an incomplete piece of the goal.
Similarly, concept drift can be a source of overfitting -- the objective you care about is the one after the concept drift occurred, but the objective you trained on is the one from before the concept drift. (Here's a scholar search for papers where the two concepts co-occur: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&as_vis... )
The problem with overfitting and inability to extrapolate out of sample is variance. The problem with Goodhart's law is bias.
I don't think a training sample is to the full population what a proxy metric is to the objective -- not theoretically, and not practically. The training sample faithfully represents the full population (by definition, if it was randomly selected). Any difference in composition is down to sampling error, and this is known from theory.
When overfitting we are still optimising for the objective, only adapting more to individual data points than desirable. Goodhart's law implies optimising for the wrong thing entirely. We have no theoretical tools to deal with this and I suspect we never will, because it's a problem of subjective judgment.