"I am given to understand that the principal absolute requirement for dividing specimens into different species is the capacity to interbreed and produce fertile offspring. If they cannot produce viable offspring, they are different species. If they can, they are the same species, though possibly of breeds having radical morphological differences."
You are in the main correct. The main definition is two populations that cannot interbreed. There are caveats to this. First, it is two populations that don't interbreed in the wild. So, sometimes two populations that could interbreed, like Darwin's finches but don't , because they are geographically, ecologically or unwilling to because of color displays etc. (Sometimes, during drought or other
environmental change, two of the finch species come back together).
Second, sexually reproducing organisms go by the above. Asexually reproducing organisms are more difficult to define. It is clear that once something is significantly different in genetic characteristics it is a different species, but there will always be arguments, especially among closely related asexual forms.
Example: If a bacteria evolves into a new population resistant to a new antibiotic, is it a new species? It could be argued either way. However, if a bacteria evolves enough to produce different symptoms, it definitely would be classified as a new species.
Since species is the lowest difference and the only one that really matters in evolution, the change does not have to be very much to qualify as being a new species. Most speciation determinations between closely related beetles or nematodes, for instance, take a great deal of technical knowledge and expertise.
thank you.