My question, specifically: why is the Fujifilm HS 10 not an SLR?
SLR - Single Lens Reflex, meaning that there is only one lens through which the viewfinder and the film are exposed to the image. This is in contrast to other types of cameras, like rangefinders, where the viewfinder was a separated optical path than the film, usually in the form of a small lens on the top-side of the body. The Reflex part comes from the rotating mirror which is used to direct the image from the lens to the viewfinder. When the shutter is released, first the mirror goes up and clears the optical path to the film.
Another characteristic of this class of cameras is interchangeable lenses. I am not sure, though, if historically all SLR's had this option, and probably some of our experienced forum members can comment on that.
Nowadays, with the DSLR technology, sensors replaced the film but otherwise the basic structure and principles remained the same.
The Fuji camera in your link is not considered an SLR for the lack of the mirror, pentaprism and optical viewfinder. There, it is an electronic viewfinder, meaning that what you see is an image generated on a tiny LCD inside the viewfinder assembly.
Definition of SLR
SLR consists of:
SLR is a term that's pretty old now, and at the time distinguished itself from other camera designs which used a separate lens assembly for an optical viewfinder, such as rangefinder cameras or twin lens reflex cameras.
Benefits and drawbacks
The Fujifilm Finepix HS10
Nowadays, there is increasingly less need for an optical viewfinder even on interchangeable lens cameras, since electronic viewfinders offer some unique benefits such as zoomed image previews, histograms etc, and their image quality is improving lessening the gap with a good ground glass screen (not that the Finepix HS10's electronic viewfinder is any good). That's why you are starting to see "mirrorless", yet through-the-lens, systems such as Micro Four Thirds and Sony Nex entering the market. They can be made smaller and with a smaller lens flange to sensor distance, the lenses (particularly wide angle lenses) can be made smaller and lighter too.
So far, all the mirrorless systems also use a smaller sensor size than full frame making DSLR (or rangefinders) the only options for those needing the characteristics of full-frame sized sensors. Also, mirrorless systems typically use contrast detect auto focus (CDAF) instead of phase detect auto focus (PDAF). Historically CDAF tends to be slower and PDAF tends to be less accurate but this is only a generalisation; CDAF is improving in the speed department and an expensive PDAF system will be accurate anyway.
To-date, the moving mirror & viewfinder mechanism that @yasp described is sort of the defining characteristic of an SLR. I think we're going to see this "clean" definition continue to erode, though, with new cameras on the horizon.
Other characteristics generally attributed to the DSLR format:
There are already Electronic Viewfinder Interchangeable Lens (EVIL) cameras such as Micro 4/3 cameras, that deliver many of the benefits of DSLR's without meeting this specific definition of DSLR, and more are on the horizon. The new Sony A55 and A33, for instance, use a translucent mirror that doesn't move, and Nikon is rumored to be introducing a "pro" mirrorless camera soon, too.
As more of these "not quite DSLR" cameras enter the market, I wouldn't be surprised to see the traditional definition of DSLR become less important, if not less clear.