Hi Rebecca! (love your name btw)
Before you read this, please know that I am not a voice therapist and am in no way qualified to help, but it really sounds like you're pinching muscles above your larynx, giving you almost a squeaky voice. Perhaps also be some tightening of the false vocal folds ( used in falsetto ) because I don't hear any overtones. It's quite mono-tonal even with the higher pitch.
To get a good sound, you really want to allow the air to flow freely through your larynx and through your nasal cavity. That's where your voice needs to resonate to sound 'full'.
Even when I fully block my nasal cavity, I don't quite get the same sound I hear in your voice, so there is more to it than this. If I really try to constrict my larynx and use my false vocal folds, I do get close. But wow that's hard to hold lol. And my poor neighbors must be wondering what I'm doing...
I kind of wonder what's happening to your larynx position actually. If you relax your voice and neck entirely, can you find your adams apple? Or feel where your larynx is? The after you're relaxed and have found it, prepare to talk. Does it move? If so, where and by how much?