i'm designing on a square grid in illustrator (made with grid tool) rotated at 45', and i'm having real trouble getting pen tool to snap accurately to the grid. I've turned off all of the "snap to" options, i've converted the grid to guides, smart guides is on. I've tried changing the snap to tolerance, and i've tried rotating the whole doc with rotate view tool, which seems to work a bit better (i'm guessing because the grid is not rotated anymore, but the document is) but this just presents it's own problems with selecting objects.Here's some screenshots, as you can see, it looks like it's aligned to the grid, but when zoomed in, it's always a bit off. Preferences shown in other screenshot too
A couple things...
I would turn on Snap to Points (View> Snap to Point)
Try using the Direct Selection Tool (White Arrow) for better snapping. You can select the entire object (rather than just a single point) by Alt/ Option clicking any anchor point.
Drag your object from the corner point rather than just anywhere on the object. This seems to snap better.
Otherwise, it could be that your fill square is not exactly the same size as your grid square (hard to tell from your screenshots).