In my state, coroners are an elected position. They require absolutely no training or diplomas even though they can declare people dead. They process paperwork for death certificates, can investigate death scenes, can seize bodies and have them sent to the state crime lab for autoposy along with other evidence of death and health, and hold a coroner's inquest which is basically a court hearing concerning the nature of a person's death (natural, suicide, accident, felonious, etc). The one from my home county attended the course in St. Louis, and he talked highly of it. However, there aren't many from the 75 counties of my state that have been to it. Even he has a couple of part-time deputy coroners to take call when he's off, but they haven't been through the training.
In many of the counties (three out of four) I've operated out of (via law enforcement career) the coroners were funeral directors that ran for office and got elected. The one from my home county actually retired from the highway department before wanting the job.
A couple of other interesting tidbits exist about them here including the fact that an arrest warrant can actually be addressed to coroners as well as police officers, marshals, constables, and sheriffs, lol. At one time, the state constitution gave only elected coroners the authority to arrest elected county sheriffs.
Also, in some states, the county sheriff is automatically the coroner as in others the sheriff also holds the title of tax collector. Here, that's not the case.
A medical examiner is a physician.