What mechanics control the path emblems on the player cards?

In Fire Emblem Fates, when you view a player card, it displays the three emblems of each path. I have seen them in numerous different states, but I cannot find anything online which explains exactly how they work. I imagine that when they are lit up (as with Birthright and Conquest below), it signifies the player has completed that path, but I am fairly certain I have seen it highlighted in different colors than the one shown. Also, how is the completion status associated between files? If I start a new game after beating one path, will my file still reflect that or must I use the Branch of Fates option on the main menu? In short, what are the mechanics that control and influence the emblems on the player card for each path?