Score annotation system?

There is no official system, per se, but generally teachers use what your teacher has used on your sheet music.

The annotations your teacher put on your music to show you errors by circling incorrect notes or something else, depending upon the context. Some teachers may underline things of interest, or circle phrases to show you the phrase more clearly. I had a teacher that used various sizes of squiggly lines to show expression. A big squiggle meant more and smaller ones less. I had another teacher circle individual notes not because I played them wrong but because he was showing me some inner voices. This is also accomplished by underlines, or even little check marks. My very first teacher used to put symbols on the music for memorizing sections. She would put in squares, triangles, flowers, etc., and then have me learn and memorize the music. Once memorized, she would interrupt me and have me play from the square in one piece. When she said that, I had to jump to that place and continue on from there. These were the fallback places as she explained in case, I had a memory slip. Other marks are used to show breathing places in voice, and also for phrase breaks on keyboard works. Little commas, slashes, and so on can mean different things.

These annotations, once you understand what they are for, make sense and more so if you practice what your teacher has shown. Sometime later, it'll all click and this all becomes natural

The finger numbers are an important part of the package but they don't make the music sing. What they do is give you which finger to use, usually as a suggestion by the editor, or implicitly by your teacher. Generally, with experience you'll develop your own fingering for passages. All of this takes time and practice. Just playing by "finger numbers" is no different than just pushing buttons.

Speaking of practice, going through the drudge in the beginning i.e., the fingering, rhythm, notes, etc., very slowly at the beginning makes the piece solid under your fingers as my old teacher used to say. This will also remove the extra circles and other mistake notations your teacher places on your music.

Once you have this part rock solid, you can then focus on the musicality. As you gain experience, you can use a combination of very slow playing, meaning even fast passages are worked on slowly for accuracy, along with the musicality help bring the package together faster, but you should always focus on the foundations first.

In some ways, I'm not sure if you've ever built a model railway/road, or put together some kits, but it's the same. For a model railway, you build the boards, put the track down then plaster and wire down along with paper mâché to make mountains and hills, and then landscape it all. This is very much the same as working on a piece of music, especially complex works. A solid foundation means a solid piece you can play on the piano with confidence.