I use Lamplight "Ultrapure", apparently same as the 2nd link but in a clear plastic bottle. I can't smell it, and my nose seems to be as good as anyone else's.
It's difficult to be precise with the terminology. All three of these are, I think, mineral oil, and probably paraffinic. The mineral oil you might get at the drug store is likely too thick, but who knows about this stuff - what's it for? In the US, "paraffin" is usually used for a hard mineral wax, which troubles us some as we wonder if that's where those thick waxy deposits came from (NB, mostly not in my own instruments), but if it's that thin at room temperature, it isn't a wax and won't become a wax. Lamp oil could be all kinds of stuff, from rendered sheep fat to alcohol, but kerosene is fairly common.
I am kind of guessing, but I think purity (as in "Ultrapure", and mineral oil is graded by purity) is connected with odor. Pure must mean "composed of hydrocarbons of a very small range of chain lengths", where impure has a wider range of molecule sizes and possibly some cyclics.