How to know whether you are an idealist or a physicalist/materialist

You probably need to be a bit more explicit in tieing your title to your text.  I'd guess you're just pointing out that these labels ("materialist" and "idealist") are both ridiculous when taken to the extreme, and that all sane people use different models for different decisions.  Oh, and that all cognition is about models and abstractions, which are always wrong (but often useful).

If I'm wrong in that, please use more words :)

As to your questions about the moon, I don't thing "observable" has ever meant only and exactly "directly viewable by the person doing the writing".  It means "inferrable from observations and experiences that are causally linked in simple/justifiable ways".

In what sense are these two viewpoints in tension?

This seems more a question of "observable by whom" vs "observable in principle."