By my understanding, a lavra is the church used by hermitical monks and is separated from society, while a sobor is more like an administrative center for the orthodox church.
comes from the Greek 'laura' (λαυρα) and referred originally to a monastery of hermits, a tradition dating back to the 5th century CE in Palestine. In the Russian Orthodox context, it continues to refer to a monastic community, frequently but not necessarily built around a cathedral or church.
is strictly Slavic, but has changed meaning somewhat. Originally it came from Old Russian suboru (субору) meaning a council, and that meaning is still used to refer to Orthodox religious councils or meetings, but it also came to mean the structure in which high church councils took place, and then a Cathedral.
In game terms, for once both Civ VI and Humankind got it right: Lavra is more appropriate for the scattered Districts in Civ VI which may be completely disconnected from the city center, while Sobor is more appropriate for the urban concentrations that Humankind
Now about them Cossacks. By the time of Catherine II the Cossacks had finally largely been subordinated to the Russian State, but they were a small percentage of the Russian cavalry, most of which were dragoons with smaller percentages of hussars and cuirassiers. The most unique thing about the Cossacks, and we'll have to wait a bit to see if Humankind
shows it since the illustration shows a musket instead, is that they were expert light cavalry Lancers and had been since the 16th century. That would make them unique in Europe by the late 18th century, since by that time most light cavalry were armed strictly with carbines and sabers. The lance came back as a preferred weapon during the Napoleonic Wars as a direct result of everybody's experiences with the Poles and the Cossacks, but in Catherine's time, the Cossacks were pretty much alone as Pig-Stickers.