Function of the Third Pin Inside Laptop Barrel Connector
Standard DC barrel female jack 3rd pin is a detect or switched pin. When the male plug is not inserted, the 3rd pin is connected with the center pin. This is used to route a signal around the jack, like a battery pack, or to signal to a controller that nothing is inserted. While I have not seen all laptops ever designed, I have seen many and I have never seen a laptop use the 3rd pin of a standard DC connector. The question you link to is talking about those connectors. Which is not the connector you are asking about.

Newer laptops that use different 3 or 4 pin cylindrical connectors are not using old standard DC connectors. These actually use the extra  pin for other signalling like power supply identification (Apple, dell, Lenovo use 1-wire or similar standards for this) or maybe dual voltage inputs. These 3 conductor connectors have a center pin, and the outer barrel is two non-connected sleeves, providing two conductors.