For me, being transgender is absolutly not "but having the body of" something. I think my body fits me very well (allthough some things are missing, but there are ways so compensate for that). Gender dysphoria, for me, is not about that.
I don't feel trapped and I certainly don't think my body belongs to another gender(I mean, it's my body so it is defined by my gender).
But I understand that not everybody feels like this and I think it's important to acknowlege different experiences. After all we are not one homogeneous group, but individuals.

I didn't know I was transgender for most of my live. I only knew that I doesn't fit. Neither with the girls nor the boys. It just felt odd thinking about me in that terms. There was allways this distance.

When I first heard the term "genderqueer", I felt something warm inside my heart. It felt right and I felt like I finally belonged.

So the thing about "wishing to be another gender" doesn't really apply to me. I am my gender and I don't wish to be another. Phrasing it like this sounds very stange to me. But that's just my experience