Eureqa has a number of metrics listed here:

The only reason that I like correlation coefficient is because I don't have to fiddle with scaling, but I suppose I can do that with any metric as long as I scale each of the input data sets by their variance?

I think I understand what you mean by distance metrics. These somehow use both X (the amplitude) and t (time) in devising an error metric. The error is not set by X alone so that slight phase shifts WRT time are not penalized so severely.

Is that right? I have been searching for something to use like that.

The only reason that I like correlation coefficient is because I don't have to fiddle with scaling, but I suppose I can do that with any metric as long as I scale each of the input data sets by their variance?

I think I understand what you mean by distance metrics. These somehow use both X (the amplitude) and t (time) in devising an error metric. The error is not set by X alone so that slight phase shifts WRT time are not penalized so severely.

Is that right? I have been searching for something to use like that.