This time I believe I've validated my polishing process... First sanding with 240 grit paper just to get rid of the anodization, then 320, 400, 600, 800, 1000, 1200, 1500, and finally 2000 (all of those wet, obviously).
Then, final refinish in two steps, sewn cotton first, non-sewn then, with two different polishing pastes. The result looks pretty nice to me (see left picture)...
The main point is to follow closely the order of the papers grit : so I kind of organized this stuff in a folder, with the sandpaper already cut ; the actual work will be much easier (see right picture).