There are two ways cameras do "digital zoom".
The first is via interpolation, they take the centre portion of the image and then scale it up by inventing new pixel values for the gaps between the actual camera pixels. This was much more common on older cameras that had limited pixel count but is still common in video where the output has to be an exact pixel size. There is little point in doing this in the field. The processing power of a desktop and the algorithms used are much better. Plus you can always undo if you don't like it, whereas if you capture an interpolated image you can't go back as you don't know which pixels are real and which made up.
The second method of digital zoom used is simple cropping. Taking the centre portion of the image and outputting this as the (smaller pixel count) image. This is done by many modern cameras with their higher pixel counts. This is perfectly analogous with cropping an image on your desktop. But again if you do the cropping on the desktop you can undo, where as if you do it on the camera you can't get the lost pixels back. Your point about setting the exposure for the cropped section is valid but if you shoot in RAW your latitude for exposure correction is much better. You might want to reframe a shot and you can't get the extra pixels that were around the image back.
One added complication is that some cameras will switch between these digital zoom methods. Cropping until it reaches a minimum pixel size and then interpolation takes over. This behaviour can usually be disabled but it can be buried in the settings.
The simple answer to all this options is to take more pictures! Take the full size picture and then zoom in (method 2), re-expose and take another picture. Interpolation in camera is rarely worth it, unless shooting video or you are using your camera as scope. You will get much better results on a computer and you can play with the settings and algorithms to get the right result.
