FOCAL LEnght and Point and Shoot Zoom

The zoom factor like e.g. the 10X you mentioned above is simply the
longest focal lengt divided by the shortest focal length of that lens.
It could be 10mm-100mm or 20mm-200mm or 100mm-1000mm. All of those
lenses would be a 10x zoom.
The zoom factor does not tell you anyting about the magnification (or
tele) factor. That can be expressed in e.g. 8x, too. This is commonly
used in e.g. binoculars where 7x is a common marine magnification, 8x or
10x a common terrestial magnification, and 15x a strong glass for e.g.
bird viewing. For photography however it is much more common to use the
focal length. Problem is that the magnification not only depends on the
focal length but also on the sensor size.

On a full-frame sensor a focal length of 50mm is considered a 'normal
lens'. A 200mm lens would have a 4x magnification, a 1000mm lens a 20x
magnification.
For a DX sensors you have to adjust the magnification factor for the
smaller sensor size by a factor of 1.5 or 1.6, depending on the actual
camera.

Ok, going back to your original question: a 200mm lens on a DX body
(Canon EOS Digital Rebel XTI) will have a magnification(!) factor of
200/50*1.5 = 6x.
And it will have a zoom(!) factor of 1 because is is not a zoom lens to
begin with.

If it were e.g. a 50-200mm lens, then the magnification factor would be
1.5x - 6x and the zoom factor 4x on a DX body.
On a full frame camera this lens would have a magnification factor of
1x-4x and still the same zoom factor of 4x.

Oh, one more note: neither the camera nor the lens determines the image
size. The image size only depends on how you print it.