Stone Scoring
Ancient Chinese Rules involved playing until no more stones could be put on the board without jeopardizing life of the groups. Then the stones are simply counted.[1] Since only stones are counted toward the score, one doesn't want to pass until all one's own territory is filled except for the two eyes required to give each group life.
In general, this imposes a two point group tax on each independent living group: The two points that are the eyes required for life don't count for the score.
But there is ancient and ancient: stone counting was the norm in China during the Ming Dynasty?. As Ancient Chinese Rules and Philosophy indicates, there is evidence that earlier scoring was territorial.
See Ancient Chinese Rules And Philosophy for more context.
(If you are looking for information about the simple counting method for Area scoring rules, which is sometimes called "Stone Counting", see Stone Counting Method.)
[1] This seems to me to underestimate the practicality of this simple method. (Assuming captures are handled sensibly) there is no need to count stones. Black and White can compare the size of their areas by alternating in putting stones in them. If White fills his up, and Black still has 3 points that he can play on, then Black has won by 3. Jonathan Reece
Velobici: This assumes that there are the same number of stones of each color, a condition that would have to be verified at the start of the game. In go clubs, bowls rarely have the standard 180 stones. This could done by counting all the stones. Ing go bowls are designed to make this trivially easy. Without these bowls, counting again becomes necessary.