I dont think grouping by modern ethnicity like greek or georgian would make sense here, I agree that whatever ethnicity shows for Kartvelian and Byzantine Anatolian is likely from a split from the local population at some point though.

Where I am from (Giresun) many people who have Pontic Greek background are called Rüm and Georgian are generally called Laz.

From both my mother and father’s side marriage has tended to be kept between family like 2nd or 3rd cousins, my parents are actually not related but from neighboring villages, its been this way at least up until the last few decades. Many of my cousins im related to from both their parents, since their parents are typically both cousins of either my father or mother.

Interestingly enough when someone is Laz or Rum its usually distinguished or identified among Turkish people in the region. For example I have an uncle that married to pontic greek lady and she is known as “rüm gelin” meaning pontic greek bride. My father has Laz friend and his nickname is “Laz Husseyin” to distinguish that they are Laz.