Oil, grease, lubricate, anoint, cream all mean to smear or treat with an oily, fatty, or greasy substance, but they vary greatly in their implications of the substance used and the purpose for which it is employed and in their idiomatic applications.
One oils the parts of a machine or mechanism subject to friction, typically by drops or squirts of a liquid substance, usually a mineral oil. Also, one oils a fabric (as cloth, silk, or paper) when one impregnates it with oil so as to make it waterproof.
One greases a thing when one rubs on or in a thick fatty substance, often an animal fat or oil, for some purpose such as to increase speed by reducing friction or as a medicinal application or as a preventive of cohesion.
One lubricates when one oils, or greases, or provides for the feeding of a lubricant (as oil or grease or graphite or a silicone) to contiguous surfaces in a machine or mechanism to make them slippery, thereby reducing friction, eliminating roughness, and preventing cohesion.
Lubricate stresses the effect intended or produced; oil and grease, the substance used or the method of its application.
One anoints the body or a part of the body when one smears it with, or rubs into it an oily or fatty substance for some purpose (as a protection from the sun or an aid in massage).
Anoint, however, is especially employed in reference to ceremonial uses of oil. In the application of oily or fatty cosmetics, especially those which are called creams, cream is the customary term.