Oil is used as a carrier, its job is to evenly spread the diamond over the lap and lubricate the lap surface.
Water is a pain to use because if you don’t get the drip just right the lap will either dry out or wash the diamond off, If it gets to dry it will put scratches in the facet. The muddy slurry your talking about gemmakerm will produce a good polish. But under a 10x loupe the facet is full of tiny grooves. Which in competition cutting is no good.
Oil is the easiest to use with diamond. I suppose any light oil will do, I use one or two drops of baby oil with 60k diamond powder on a BATT lap.
Rubbing the mixture onto the lap with my finger and wiping any excess off, till the surface of the lap feels almost dry.