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I fully understand the confusion about impedance matching. There is a huge amount of lore on the internet about audio impedance matching. It mainly goes back to the days when transformers were used between audio stages in a typical circuit. That is now relegated to history.
A transformer coupled stage requires careful impedance matching between stages. Transformers are not the preferred option in modern design.
So, what really matters is the voltage level produced by the computer audio output, whether it be headphone or line out. When the destination load it 10k ohms, the voltage at the destination will be essentially the same regardless of the driver source resistance of 10 or 200 Ohms.
There is no information contained in the AC’97 specification that would demonstrate that the headphone out is any different from the line out when driving a 10k ohm or higher load.