Keep in mind that the analysis was probably one of the earlier ones written. If it were written today by someone with a more up to date understanding of the theory, it might be a lot more clear.
That said, I want to address the idea that “rescuing the princess” “really REALLY feels like obtaining.”
Rescuing the Princess without context can be Doing or Obtaining. Or anything. It’s easy to see a word like “rescue” and think “oh, they’re going to GET her, to POSSESS her, so it must be Obtaining”, but “rescue” is an activity just like “walking”. And if they were walking with the princess, it might really really feel like Doing. But again, without context, “walking” could be anything. After all, one can achieve “walking with the princess” just the same as they can achieve “rescuing the princess”, just as one must engage in either in order to achieve them.
So the difference between rescuing the princess as Doing vs Obtaining, then, comes down to whether conflict arises from engaging in or achieving rescue.
I haven’t seen Star Wars since at least 2017, so forgive me if I get some of the details wrong. But as previously mentioned, the fights they engage in aren’t to stop the rebels from taking the princess. The empire let’s them take her. When Luke and Han dress up like Stormtroopers and pretend to be transporting Chewbacca, they don’t get stopped and then have to fight with the bad guys because they are achieving rescue, but because they are engaged in it, because they are trying to gain access to an area they are not supposed to be in. When Obi-Wan runs across Darth Vader and has to sacrifice himself as Luke and friends watch, it’s not because he is achieving rescue, but because he is engaged in it. He is just walking through the hall when he comes across Vader.
To make this scene more about Obtaining, you might add a bit of dialogue where Vader tells one of the officers to be on the lookout for a plot to rescue the princess and to make sure no one gets to her. With that one line, the whole scene could potentially be about who possesses the princess rather than stopping rebels from sneaking aboard and following rebels back to their base. Or you could have Vader refuse to let them go with the princess. Instead of a plan to follow them back to base, he says to follow them until they can get the princess back unharmed. Just something small to shift the focus slightly is all it would take.