Hi Anders,
In the way you describe it, it is not really how it works.
What you have rotated was the selection based axis. What you need to rotate is the Object axis. (If the lid is an object on its own).
There are many ways to open a lid with points move only. PoseMorph, while using Rotational and define a specific axis, or using a Cluster Object and use a parent for the points as the axis.
Scene file and screen capture
https://www.amazon.com/clouddrive/share/E4BLuWk4uwq0nryX6WHRALKOXEs5ELAGAoq5T7XAywu
What you have experienced, and I guess here, is the movement from one point position to the next, a straight line. It doesn’t matter how the points where moved; what matters is the information stored in the keys.
Since I consider guessing as a pretty nonprofessional behavior, I love to see a scene file, reduced to the problem typically. Images usually are not really at all helpful. Since we live in ransomware times, I use ONLY DropBox, Amazon, Adobe, Google, Wetransfer, or Apple as cloud service, with no abbreviated URLs. I can always set up an upload link if needed.
(Edit: With not professional, I mean that dumping many questions and guesses into the forum would be unprofessional from my side. I find platforms with that kind of stuff in it super hard to read, challenging to follow, and wasting everyone’s time. Instead, I offer to look at the file, and when I get a reproducible solution, I post the answer and perhaps some alternatives. Besides, I started a few years ago to make 60-second clips, “running” through a setup, to visualize the setup. (Nearly 1300 clips by now.) It seems to work. Otherwise, the communication might be extended. Sorry, it was never meant to characterize nor evaluate the question or the one who asks itself. Not at all! Really sorry. Well, now I took a lot of time. I hope it is OK./edit)
Of course, to attach a zipped file to the post is the easiest way.
All the best