What on a higher level should I be thinking about as I play through a piece such as Bach’s Minuet in G major or his Minuet in G minor?
Once you have the muscle memory of a piece down, there's an important component of music that can often be overlooked by musicians who like to think: the emotion.
One good goal for learning a piece of music is to get it to the point where the playing of it is so automatic that you transcend the concept of playing it correctly and are able to focus on playing it with feeling. Listeners want to feel something, they (almost all of them) are not looking at how many mistakes you make or whether you understand structure, forms, or theory. They want to feel something.
Each piece of music (in any genre) has one or more emotions "encoded" into its composition, and when a piece is performed, the musicians can, and usually should, both interpret the inherent emotions and also communicate their own take and feelings that they are inspired to feel when they hear or play the piece.
So when you have mastered a piece in the technical sense, the next step is to think about how the piece makes you feel, what it reminds you of, what kind of emotional conversation is started inside you when you play it or hear it. Then, your goal is to continue that conversation. Some musicians attempt to relay their best guess at the original intended emotion(s) only, others add a little or a lot of their own take on things, and some even work to subvert the original intent and use the raw material to convey something very different.
When you are engaging in an emotional conversation with the composer and the audience, that is when you are truly making music.