"Damping grease" is the proper name for what you are looking for
http://www.nyelubricants.com/lubenot...rease-03-1.pdf There not being a large base of US lens makers most of the literature on selecting lens helicoid grease is in Japanese and Chinese. But the greases are the same as the US product.
High vacuum grease is another alternative - though not exactly a commodity item.It won't migrate no matter what - really great stuff and a lot easier to clean up after compared to some of the stickier damping greases.
The 'faucet grease' sounds like it belongs in the damping grease camp. Microtools is repacking either Nye or Corning [or Gongzhu Optical Product and Tractor Works] grease. I didn't look at the pricing differentials but a small amount of the stuff will last you a lifetime.
Any sort of household grease will separate and you will get oil all over the inside of the lens.