In microscopy, the term wide field is often used to describe lenses that increase your field of view when looking through a microscope. I haven't heard the term used in photography, but have heard wide angle used frequently. Do they have the same meaning? If not, what's the difference? If the same, what's the reason for the different terminology.
Technically, Field is a specific linear dimension measured perpendicular to the optical axis of the lens system. The width of a field is determined by the angle of view (AoV) combined with the distance at which the field is measured.
For things such as microscopes, this is fairly straightforward because the slide upon which a specimen is mounted is usually at a fixed position. For binoculars, "Field of View" is normally described in terms of how much linear width will be visible at a distance of 1,000 yards (3,000 feet or 914.4 meters) from the binoculars.
With cameras, we do not tend to shoot our subjects at the same distance. So rather than calculate a linear width for a specific angle of view at a specific distance, we just talk about the angle of view itself. Or we talk about a specific focal length that will yield a specific angle of view with a specific format size. Format is defined by the width and height of an exposed portion of film or image sensor.
Because things such as microscopes and binoculars tend to magnify things significantly, the angle of view that a wide field microscope or wide field pair of binoculars is still fairly narrow compared to what we refer to as wide angle camera lenses.
For instance, a wide field pair of binoculars may give a field of 420 feet at 1,000 yards. That figures out to an 8° AoV. On a full frame camera, an 8° horizontal AoV is yielded by an approximately 260mm focal length. On a 1.5X APS-C camera, that same 8° AoV requires a roughly 180mm focal length.
We consider such lenses to be narrow angle of view telephoto lenses.
What we consider "wide angle" lenses are those that provide around a 65°-70° or wider angle of view. In other words a 28mm-24mm or shorter focal length on full frame, and about 16-17mm or shorter on APS-C cameras.