From my NFPA Fire Protection Handbook:
1. Tagliabue "Tag" Closed Tester for liquids with FPs at or below 200F (ASTM D56).
2. Pensky-Martens Closed Tester for liquids with FPs over 200F (ASTM D93).
3. ASTM D3278 (apparently a closed test) is for finishes (paints, enamels, laquers, etc.) with FPs from 32F-230F .
4. Open cup tests (Tag Open Cup - ASTM D1310, and Cleveland Open Cup - ASTM D92) are sometimes used for flammable liquids "in transportation." Open cup flashpoints represent conditions with the liquid in the open and are generally higher than closed cup flashpoint figures for the same substances.
5. For motor oils, the standard consistently used (at least in the U.S.) appears to be COC/ASTM D92.