Coconut water is the liquid in a coconut when you harvest it. Coconut milk is a thick white liquid made from the moist white pulp in the coconut and has a higher concentration of nutrients, electrolytes, n' whatnot than coconut water.
With regards to your pup, I would highly recommend that you do not remove it from the mother plant. The rapid growth signifies that the cactus is diverting nutrients from the cutting to help grow the pup (a pup on my bridgesii appeared two months ago and is now ~6-7 inches high and still growing strong). The nutrient-rich stock is propelling your pup to new heights far faster than it would ever have a hope of reaching if it's mama wasn't so nurturing and separating the two would be a grave error IMO.
I could say many many many many things about why petroleum products are terrible to use around plants, the main one being that many are toxic to plants and may kill them if ingested. To encourage more pupping, I would suggest increasing access to water, increasing sun exposure, and fertilizing the soil with Miracle Grow and a coconut milk dilution as mentioned before sound like good ways to help encourage further pupping.
However, I wouldn't try to get as many pups to sprout as possible as soon as possible as rash and impatient actions and intentions tend to be the leading cause of premature plant death. I'd suggest you tend to the cactus as you have before, watering occasionally, to help support the growth of the pup that has already emerged. I would rather have one pup shooting up like a Saturn V rocket than have a dozen pups sprinkled around the cutting growing as slow as lophs do.
Hope these tidbits of advice help,
-Sherdan1