The good news is that the horse seems to appreciate the quiet way you have with her over your SIL's manner. I still, however, think she has something going on with her poll. She wasn't pulling back when you were petting her there, but the way she kept shaking her ears the entire time you were doing that suggests to me that something isn't quite right. I also think she's uncomfortable behind, as sometimes she did not track up evenly with the left hind.
The horse needs lots of steady, quiet handling. They should be able to accidentally knock a pole with their feet and not spook. But you persist with her and she gets better, so this is good.
I would be very reluctant to chase her around a round pen at this stage. I can see why she gets panicked. In the video, when you used the lead to put a bit of pressure behind her to get her to circle around you, she SCOOTS and her head flies up and she looks a bit worried. Now, you want the horse to move off pressure and be light to the aids, but you don't want them to be frightened of the pressure and dart off like you've zapped them with a cattle prod. They should be responsive to the aids, but relaxed and accepting of them. A horse who seems this spooky about pressure behind her can very easily panic in a round pen, when she's going to feel trapped. The nature of the round pen is that they can't escape the pressure, so the burden is very much on the handler to think about their body language and be in control and aware of how each and every move they make effects the horse. That is not to say round-penning a horse like this can't be done or be beneficial. An experienced trainer could probably use the round pen/join up situation to help this horse through her concerns and show her how to accept the pressure. But if you're new to round penning yourself and haven't developed spot-on timing and feel, you probably should leave the round-penning aside for the moment.
I looked at your other thread and agree with Speed Racer's comments on that one. In order to be a functional equine citizen in the world, the horse needs to accept haltering, bridling, shoeing, grooming, etc. etc. outside of her stall. I hope the trainer coming on Tuesday is the kind of trainer who this horse needs. Someone who's going to smack the horse around when she "misbehaves" like your SIL was doing in that video is going to confirm to the horse that people are bad news. A gentle, quietly firm and kind trainer will do you and this horse a world of good.
The horse isn't ideal for a beginner --but she seems to like you and you have a good way with her. With the right sort of trainer, I think you can make progress with her.
And vet and physio are certainly in order as well.