To me her catching is not so much a trust issue as a bad habit. She is in no way frightened of you in these videos.
You have 2 approaches you can try. As FF said you must take a leadtope in with you every time you go.in the field even if its just to give her the carrot, because if you only take it when you want to get her in she will know that's what's happening.
First off in the first video before she heads off trotting you had tufned your back and were backing up to her. She didnt actually move away until you then turned to face her, then you took a step back when she moved away.
So the backing up to her was working and she was curious. You needed to stick at backing up to her all ghe way up to her give her the carrot ans slip the rope over her neck as you feed the carrot.
You should only have turned to face her if she had gone to walk/trot off. You can forgive one or two steps but any further and then you need to take charge straight away.
If she does walk or trot off, the not letting her stop is brilliant and a huge well done for that because in yhe videos where you are filming, you are obviously walking with much more purpose towards her and that's what's tired her out quicker and why she gave up, in the first you were kind of wandering along with her while she free lunged around you lol.
What you now need to do is really take charge, start dictating her direction, so if she moves left, send her right and vice versa. In the end you will get a pony that looks like its dancing lol.
You will be required to run at first, and twirling the leadrope is necessary to help send them.back again, and its a lot of to-ing and fro-ing to start, but in the end she will get to learn you can direct her by side stepping across her path, and then eventually if she moves left you can literally take a step to the right and send her back which results in the end with a pony who just stands and gives up.
I wish we had videoed us with Stella, vecause this was exactly what we went through with her and in the end she just used to sway left or right to check and see if we were on the ball and if she could get away with running off or if we were just going to counteract everything she did lol.