To reduce the graticule to 1px lines, right after you delete the white, nudge the selection up 1px and delete, then nudge 1px to the left and delete, then nudge it 1px down and delete.

Also, if you set the default interpolator to Nearest Neighbor, you can skip the posterization step. Preferences > General, Image Interpolation is the second option from the top. I believe it's set to Bicubic by default.