I am so confused by the way HD is labeled these days.

The Canon HV20 is labeled as a 1080i camera, which is 1080 vertical interlaced lines, correct? Canon also claims that this camera is also capable of 24p. Isn't 24p 24 progressive frames?

If that is the case, than the question that has been making my head spin for a long time is...isn't that 1080P since it's 1080 lines of progressive frames? Why not just call it a 1080p camera?

That because the 24p is laid into a 60i stream. You actually have to do something to the footage to convert it to pure 24p, like capture or convert in NeoHDV.