I have an advantage as I prepared a spreadsheet and so could see how the heir and house thing worked.
The first heir is born in the first house. He fulfils all the moving requirements and moves to the second house, where the second heir is born. Second heir completes all the moving requirements and moves to the third house, where the third heir is born... etc.
So, the tenth heir wlll be born in the tenth house. The game ends when he has fulfilled all the moving requirements but before he moves to the eleventh house.
I can understand the confusion because I too get confused with generations and to whether or not the founder of a legacy is the first generation or no generation. In my head, I always think that the first generation begins with the birth of the first child but, from what I've read of the legacy stories elsewhere and how they are numbered, other people don't.