Repairs are built up in the damaged area and then refined level with the surrounding stone, so the repair is finished as part of the surface rather than sitting on it.
Materials are matched to the stone and to the location, tinted against the actual piece. On patterned stone this is done in stages rather than as a single fill.
Lippage correction removes material from the high side. How much is available depends on tile thickness, and where it is insufficient the honest answer is that the height difference can be reduced rather than eliminated.
Voids in travertine and other naturally holed stone are filled where the finish requires it, which is normal for that material rather than a fault being covered up.