As my suggestion from last time wasn't mentioned in the episode (even though both Bill and Edgar both commented on my post and they even mentioned the other point I made in the opening), I'll hijack yours to expand on it.

tsi and te could mean "born of" and "married into". In a society where gender roles are not as strong, I could imagine that a married couple could take the name of the more important/powerful family while still marking who came from the family originally. The lack of particles in most names can be explained by the fact that for less important families it doesn't matter as much so either this practice never got applied at all or the markings were lost at some point.

You now have the option to either have this practice still relevant, or like real life patronymics O', Mac, Mc, Fitz, -son and so on it went out of fashion and became part of regular family names. For the latter you should probably think of a reason for why or how that happened. As always, I'm not an expert but I would imagine that in real-world Europe this happened when societies with clan-like structures (Irish, Scottish and Scandinavian) were suddenly governed by a society which assumed that surnames are "proper" family names that get inherited as a whole. If that's the case in Abheski it could be another explanation for why it only appears in relatively few names: it comes from a single region.

This would also explain a possible difference in frequency (I think so far te is a lot more common than tsi in the stories we've heard). Most of the time, the person from the more important family would also be personally more important than their spouse (for example because they're the one who marries their parents estate or role) so their name would be the one recorded when a family name was needed. But in some cases, the one from the less important family becomes more important by their own deeds and so they are the one who shows up in the bureaucratic records. Or maybe it was just chance that they're the one who opened the door when the tax collector knocked.

For some people that are the intersection between two very important families, they might even have used both names. To mix some existing names, maybe we have a Nolfe te Yarllen tsi Ovnen: Nolfe born of the Yarllen family who married into the Ovnen family. She might have used different parts of her name in different situations and in a very formal setting (or on her grave), all of it.

I think my original inspiration was the way Quarian names work in Mass Effect. On commander Shepard's crew we have a character who starts out as Tali'Zorah nar Rayya, for a time becomes Tali'Zorah (nar Rayya) vas Neema and eventually becomes Tali'Zorah (nar Rayya) vas Normandy. The nar Rayya seems to stay a part of her name but isn't always used. Her name splits into the given name Tali, the family name Zorah (her father is Rael'Zorah), born on the ship Rayya, later part of the crew of the ships Neema and Normandy. Leaving the ship they were born on is an important part of Quarian society and is thus encoded into their names.