I fear you won't get a very satisfying answer to this question because what you are asking us to do is something historians and anthropologists largely abandoned a long time ago.

While early anthropologists were very concerned with setting up typologies of cultures, even compiling "trait lists" of what made a culture that culture specifically. However, these approaches have been pretty much abandoned in modern anthropology and history because they were very hard to defend as "natural" categories, rather than largely arbitrary divisions.

The critique of these approaches argues that they are prime examples of cultural "essentialism", or the proposition that certain things make a culture specifically that culture or otherwise define the "essence" (hence "essentialism") of that culture. However, modern anthropology has really moved away from this view that culture is a thing in itself or a quality that a person or society can posses and towards the model that culture is something you do.

That may sound like splitting hairs, but it is important given that cultures are constantly changing and very rarely are they entirely discrete, blending into each other at the edges and with many subcultures dividing up the space within a culture. This is because culture is something that people do, and people even within what we could call the same culture have very different social and environmental contexts, historical contexts, and individual dispositions that means people with "do" the same culture in very different ways, especially over time.

The point of all this is that delineating the boundary of any single culture, much less an entire culture group, is going to be a tricky proposition and very much dependent on the criteria you choose.

For instance, being part of a single or related speech communities is one method for grouping cultures that is quite common, but has its pitfalls. If you take this approach you may get some fairly "intuitive" or "common sense" groupings (say, Scandinavian nations grouped together), but you also get some weirdness such as grouping Romanian and French speakers together in the same culture group.

That isn't even considering how cultures change over time. For example, how much Irish culture is similar to English culture in 1444 compared to 1821 is going to be difficult to account for unless you change your criteria or your groups over time.

In summary, cultures are not discrete entities (at any one moment, and especially not through time) and so any criteria you use to define culture groups are going to be largely arbitrary and the groups you make will be different depending on the criteria you pick.

While it would be very difficult for you to get a satisfactory answer from a historian or anthropologist since this kind of classification isn't really part of those fields anymore, if you did want an answer you would need to start by settling on a set of consistent criteria for grouping together cultures. E.g. what does it mean to be part of the "Byzantine group": is it a shared language? Adherence to a Byzantine legal system? Acknowledgement of the the Emperor and Patriarch in Constantinople as heads of state and church? You get the idea. You could interpret these groupings you've listed in any number of ways, so you would need to settle on a consistent way to interpret them before anyone can give you a satisfactory answer.

I can only answer the questions about the Mongol cultures but I hope I can help you with that. Buryats are considered to be mongol and that long ago they were even more closely related to the Mongols in term of culture. James Forsyth makes it clear in his book A History of the Peoples of Siberia that the Buryats were a group of Mongols, calling them "Buryat Mongols".

"These grasslands east of [lake Baikal] were the home of pastoral nomads from time immemorial, and indeed it was on the [Onon river], just south of the present day Russian border, that Chingis Khan was born. There was therefore some justification for the Buryats who lived to the east of Baikal considering themselves to be 'pure Mongols'"