Interesting questions! I'm not sure if I can answer your questions fully, but I'll share what I know and have found.
Is the term Ellerman bombs really widely used?
In the field of heliophysics, the phrase "Ellerman bomb" is more or less widely used, in the sense that various approaches to understanding the phenomenon use the term. For some examples, there are studies that try to interpret observed solar activity as Ellerman bombs (EBs): here studying magnetic reconnection, and here studying UV bursts, and here they interpret the "rapidly flickering, flame-like features that appear intermittently in the wings of the Balmer Hα line... that outline the magnetic network" as EBs. Just a few examples among a rich literature that is still active and has been for decades. And here where they study the dynamics of EBs in the context of sunspot dynamics and they observe three EBs.
I found that sometimes they are referred to as "moustaches" of sunspots ;) (e.g. see the conclusion section of here.)
Is there a commonly accepted definition?
Yes. The second paper cited above has a brief historical overview of Ellerman bombs in their section 3.6. In the first paper cited above, the precise definition of the Ellerman bomb is given: "Ellerman bombs were first observed by Ellerman (1917) and are small-scale (1′′ or less), short-lived (2–15 minutes), impulsive events detected in the lower solar atmosphere."
The current paradigm for understanding the emergence of EBs (and sunspots and other solar surface phenomena in general) is magnetic reconnection in the solar atmosphere, which was greatly bolstered by the advent of direct imaging techniques, e.g., the Hi-C observations and subsequent studies (circa 2012).
Also: How exactly have their locations been identified in the picture above?
This is a technical question. First, usually the H$\alpha$ spectral line of the region of interest of the Sun is checked, and EBs are identified as bright features in the wings of the H$\alpha$ spectral line. (The spectral region within the halfwidth (FWHM) is called the "kernel" of the line, and the regions outside are the line "wings".)
More fully, time sequences of H$\alpha$ and infrared are used to obtain magnetic field information of the region of interest, and other information, e.g., polarimetry, are obtained. Then correlations between the magnetic field structure and polarimetry are searched for, which, if found, can be used to more precisely identify the mini-sunspots known as EBs. At the spaceweather.com site the image in the OP is from, they provide another image that helps illustrate this:
In trying to explain the EBs, they are usually explained as points of magnetic reconnection near a sunspot that are less intense and smaller in extent than the sunspot. Typically, i.e. the abstract of this, the EBs are observed to form in magnetic field configurations with close and opposite polarimetry, which is suspected to be a point of magnetic reconnection in the low solar atmosphere.