Actually, the Yedinovertsy strictly speaking came over from the old rite to the official Church around 1800, after the Church and this group of Old Ritualists agreed in principle that the two groups were not separated on matters of faith, but only ritual. However, the Church had not yet lifted the anathemas imposed on the old rite in the wake of the Raskol rebellion. So the Church hierarchy was not as accommodating to the yedinovertsy as they should have been, and the latter had to make several changes to conform to the Church's standard. For this reason, many Old Ritualists continued to live separate from the Church.
The Erie group came to the Russian Church Abroad much more recently, after the ROCA synod officially lifted the anathemas on the old rite in 1971. Up until then, they had a separate hierarchy from the Church.
Old Believer is the term preferred by the Old Ritualists themselves, because unlike the official Church, they believe that their differences in ritual do constitute differences in faith. There is no meaningful distinction between Old Believers and Old Ritualists within that group.
There is an important distinction, however, between the "popovtsy", Old Ritualists with their own hierarchy, and "bezpopovtsy", or "priestless", who believe that the whole hierarchy fell away from the faith under Patriarch Nikon, and with it, apostolic succession.