Ah, I see I was incorrect above about the copyright thing. The copyright line in this theme behaves differently than another theme I an using (regulus).
I dropped the header.jpg into the theme folder, and it worked as expected. It would be nice if there were exact dimensions for the header area and the header.jpg completely filled the header area, no space around. Thus retaining the h1 tag.
I tried the header-background.jpg, and stopped fooling with it after several attempts to make it completely fill the entire header area, using the same image as in the header.jpg file. I tried varying the size, and I could never fill the space to the right of the image, was always a space there. Also with some sizes, the image was cut off at the bottom. But then again, I was probably misusing the header-background.jpg file in the first place, I don’t know.
In any event, I am MOSTLY satisfied, only if the header.jpg with the right dimensions would exactly fill the header area. ??