Can you provide some sources that suggest China's claim was finger 8? If that's the case why would China dispute and claim up to finger 4? The issue would have been settled there and then since India's claims didn't go beyond finger 8 as far as I know.
That is because we Chinese are the bully here !
After we secured finger 8, we want to push the indians around more !
Also, by pushing the indians beyond finger 4 will deny them access to fresh water.
The water from finger 4 and beyond are salty and can not be used for drinking. This way, the indians will not be able to deploy large amount of troops near by !
Now as far as the Chinese claim of finger 8. This was the original 1960 demand, after we destroyed the indians in the 1962 war, we have been patrolling it there until the 1980's when the indians slowly creeped in and we pushed them back again to finger 8 around 2010. Now we got even more from it.
You have to remember that, now india is in a world of HURT ! They are in no position to negotiate both, economically and from a superior position point of view. Please see the posts above by me. Now you understand why the indians are so desperate to end this stand off by giving concessions to Chinese demand while playing their typical media mental masturbation's victory; while the reality keep slapping them in the face !