"It appears that her family arrived Aug 11 1874 on the Britannic. It has a Mary Greenwood listed as being 17.
Should I follow through with this lead? How do I know it is the correct direction? "
An important element of research is to take all possible roads, and being prepared that some are going to dead-end. Ruling-out is as important as ruling-in.
You mention this Mary Greenwood arrived with family, correct? Researching those members (where did they end up?) might answer whether or not you're on the right track.
Were you able to turn up any bap records of her children? These might have associated family members listed as godparents, as well as more docs with her maiden name. You can never have too many docs.