There are a lot of oil seal types. Piston seals are also referred to as  radial seals.  This is where the seal is fitted on a shaft with the sealing lip in contact with the housing bore. V rings are also known as external lip seals.  A symmetric seal would be symmetrical and should work just as well as a rod or a piston seal would.  An axial seal will seal axially against a machine component or a housing.

The type of seals used  for a grease seal or an oil seal could be oil seals or really any other types.  An oil seal is also sometimes called a  flexible lip, a radial lip, or a rotary shaft seal. Grease seals and oils seals are used  to help  retain or sometimes separate  fluids or other lubricants.  In most cases, these seals will be used for rotary applications.