Maybe it's just because I suck in English, but I can't seem to fully grasp what this bullet in the Hunted Down mission briefing is supposed to mean:

While carrying the data chip, a hero or Han Solo can interact with the terminal (tech) to upload the docking codes.

Does this mean that only the figure carrying the data chip can perform the tech attribute test on the terminal? Because we played it that any hero or Han Solo can perform the test, regardless of which healthy Rebel figure is carrying the data chip (specifically, Gaarkhan was carrying the data chip but Mak performed the test).

That text means that the hero that is carrying the data chip is the one that has to perform the attribute test; unless Han Solo is carrying the code and he passes the test automatically.

You are also unable to pass the data chip so you need to be decide who is picking up the codes and who will be going to finish the mission.

Indeed, only the Hero or Han Solo, the figure with the actual chip, can interact with the terminal, uploading the codes from the chip to unlock the docking clamp.

As Deranged says above, you cannot pass the chip, so the figure that picks it up will likely have to be the one to do the interact with Mechanics. However, if the figure carrying the chip is wounded, they drop the chip.