Ferdinand Geber has done a lot of research about the family Rivinius. I have a whole file inheritad from my father that contains the information that Geber has collected until about 1980.
The name Rivinius descents from a man who came from a village called "Rivenich". When he went to Trier, he used the name of his hometown as his name. Later his descendants changed the name from "Rivenich" to Rivinius. (see my post to "Rivinius-Rivenich") The name "Rivinius" has nothing to do with "Bachmann". The Bachmann-descendant translated the name to Rivinus. (The third "i" is missing.) About the Rivinius-line that went to russia I have the following information: Jacob Rivinius (15.08.1771) and his wife Juliana Johann Karcher went to Grosliebenthal in Kaukasien in 1819.
Jacob was the son of Johann Gregorius Rivinius who lived in Enzberg (Baden Württemberg / Germany).
They had 11 children, a lot of them died very young. Two of the children, Matthäus and Jacob Rivinius came to Grossliebental and went later on to "Bessarabien".
Matthäus had a son called Gregor. This Gregor was the father of Johannes and Gottfried, who both emigrated in 1900 to South-Dakota!