Head bobbing at the trot can be a big lameness or pain sign and if she does it all the time it might be worth looking into. It's not the difference between "collecting" or "head setting" and not. Maybe there is something else going on that the previous owner was hiding behind "lunging issues" with. I hope not, please don't mind me or be upset if I'm wrong I could just be projecting my own present experience.
She is so pretty though! I really like her markings, dark surrounding the white!