Hi AJ, how are you doing? How is the knee?
I had a similar situation with my knee, and this is what I learned. I am not much of an anatomy expert, but it may be of use to you.
In the seated hip openers, like agnistambhasana, what should be happening is that the femur should be rotating externally in the hip socket. When this happens sufficiently, the legs rest easily on top of each other. The problem I experienced (for me, in pigeon pose) was that the femur was not rotating that much. By “going deeper” into the pose, I was creating flexion at the knee joint, which stretched the outside of the knee, eventually creating pain like you described. I also experienced the poses becoming easier and eventually had a similar popping sensation. Preceding the eventually problematic pop, there were some very minor pops that seemed to not cause much harm as far as I could tell at the time and which I largely ignored (not the best decision). I experienced a similar pain sensation around that area near the knee in seated postures like sukhasana, which also require a good amount of hip rotation. For me, it really all seemed OK (or at least I could believe that) until it really wasn’t OK at all. At that point, it took a few months to heal and get back to a place where I felt more comfortable with the knee again, especially in hip openers.
The pictures you posted show the knee being pushed down. As someone having had this experience, I do not see how this would create the desired rotation of the femur without also creating flexion at the knee joint, particularly expansion at the outside of the knee joint that would stretch the ligament there (though some may have sturdier knees than me). Stretching the ligaments around the knee is not desirable to do even if it may produce a stretching sensation in the hip. Ligaments are not very stretchy and heal much more slowly than muscles.
In my case, after the pop, I had to back way off of hip openers, pretty much doing no hip openers for some time, then only very gentle eye of the needle with support of a wall, then advancing to more poses with caution. I paid very close attention to any sensation around the knee during hip openers and other seated postures and backed off if there was any (which was of course was what my teachers had always advised). At the same time, I focused on making sure the knee was aligned in all poses, activating and strengthening the quads (particularly the vastus medialis) to help stabilize the area, and stretching the IT bands (which were tight on me and also creating tension around the knee). Your body may have different issues. Probably needless to say, my hips lost a lot of flexibility during this recovery time as there were few ways to stretch them without creating pain/injury around the knee. The flexibility in the hips is coming back.
I am now more careful with pain on the outside of the knee and more careful to distinguish in general between pain and soreness from stretching or strength work. I feel that had I backed off and resolved the issue earlier, the recovery would have been quicker and I would have made more real progress over that time period. Pushing onward was not worth it at all. As it was, I learned something about recognizing sensations in the body, respecting the body (especially the knees!), ahimsa, non-forcing, patience, caring/compassion, gratitude and other things. These were great lessons, but I think maybe it is better to learn them without injury if possible. Yoga has a great ability to heal when practiced mindfully, but is perhaps equally powerful when practiced unmindfully.
Best wishes and happy practicing! Namaste.