Thanks, Sucom. I have a feeling there was more to it.

I came to piano lessons after having had violin lessons as an adult student. I was very interested in practice approaches and any tools for learning to play, and my piano teacher stresses both of these things. What I've gotten is an element that comes
before what you describe. The OP's history reminded me of my own: I am musical so I could dash off early music without needing a practice approach so when the music got advanced I didn't have one.
So here, roughly, is what I've learned for
approaching a piece - which has been invaluable for me. Some of it I came up with:
- Study the piece. Notice if it has an ABA form, maybe modulating the the Dominant key and back, and generally its shape. When you work on a section, know what chords you'll be playing. Have an idea about your piece before starting it.
- Divide the piece into smaller sections, and those sections into even smaller bits. Say you have a section of 10 measures, a small bit might be a measure more or less. Think of the hardest section as "number 1" (to be practised first).
- Work out fingering. Check with your teacher about the fingering. Or have your teacher propose fingering. If something isn't working, suspect the fingering and check with your teacher again if needed. Stick with that fingering from then on.
- Now take your "section 1" (the hardest part of the music) and start with the
last measure. Get the fingering into your hands, and the motion. You might start with the last beat of the last measure if it's hard, and you might ignoring counting at this point, if it's hard in any way. You're going for the right notes with comfortable hands. Say you've done measure 40. Now work with measure 39 and do the same thing. Do an overlap - m. 39 into part of 40, maybe starting with the last beat of m. 38. Then join m. 39 and 40 together. You will always be playing toward what is familiar which you've practised for "correct notes/fingering". Keep going until you've done the whole section. Then work on another section for "basic notes" in the same way.
This is the first "layer" --- working toward the right notes and fingering with good basic hand motion. You should end up not having wrong notes happen.
- Add a layer to this: maybe fine tuning timing if you had to work ultra-basically only on "right notes/fingering" if a section was hard --- not a tempo --- eventually a tempo. If you lose the notes, go back to this where they get lost.
-- Add the next layer: dynamics, rubato if it fits, whatever to make it expressive.
Here you are building the piece from the ground up in small sections and in layers. The principle is that we can only concentrate on one new thing at a time. For a professional musician, many of these things are instant and might happen in one practice session. We students are creating practice habits for the first time. If we are given this as beginners, then it's part of us as we get to advanced music. If we begin as adults there is a great danger that it will be missed, and we're in a muddle of trying to "do it all" at the same time. Even if we work on a small section, trying to get right notes, expression, timing, both hands ---- all in one shot --- while we also don't have good technique as part of us yet --- stay stuck and we can't refine the piece.
I have found these things to be very powerful. It seems a very slow way of working in the first stages, but eventually the piece comes together much faster. At least that's been my experience.