Thanks for those illustrations Tom!
My methods are similar with only a slight variation.
I construct the eye 'pegs' by hand, using a file and just work my way around a small narrow cylider of whatever material is being used (after first whittling it roughly cylindrical with a knife). My pegs don't taper- I think that straight sides with inlay will hold better in pockets in the long run.
Not all eyes have to be circular- hand tools allow you to create ovals and elipses (for when you want the eyelids to be half-open revealing only a portion of the eye)
My one concession to power tools has been to use them to carve the pockets into which the inlay material will fit. I can get nice straight sides and flat bottoms to the pockets, for a secure fit.
For the amber eyes (like in the frog ojime), a peg of amber was generated. At the end of the peg, I carved a concave recess. This was painted with sumi ink. The end of the peg was then gold-leafed. This composite peg was then inserted into the pocket, cut off after the glue had dried with a fine jeweler's saw, and then polished and rounded off to eyeball shape. With transparent materials, always polish the end and sides first; then insert. With opaque, this step isn't needed.
A funny note- I worked for 1 1/2 years trying to shape tiny eyes about the size of a coriander seed in my fingers, until I read about the peg method
For irregularly shaped inlays, as Tom suggested, the methods can be quite different. One main difference for me is that with round inlays (eyes, spots, warts) I cut the pocket first and shape the inlay to fit. With irregular inlay, I shape the inlay first, and then the pocket to fit.
-Doug