I believe the term “diamond extender” originated with the use of diamond paste. Diamond paste is a diamond and grease compound that used to be popularly used on canvas and resin pads/belts, and on wooden or composition spool polishers, and on Crystalite “Star Laps”, etc.. It was used mostly in pre-polishing and polishing cabs but was also used some in faceting with laps like the old phenolic laps. The “diamond extender” fluid was/is just lite oil used to thin the grease in the diamond paste which allowed the diamond paste to be spread more easily and evenly. Crystalite used to sell a diamond paste kit that had several syringes of color coded (by mesh) diamond pastes and a bottle of their trade marked diamond extender called Crystalube which was a silicon oil product. But many thin oils can be substituted. In addition to the ones Alan mentioned a lot of folks used to use sewing machine oil.
Larry Cashatt