In the end the concept of a species is a man-made construct that tries to shoehorn the evolutionary diversity into pigeonholes. For example, your definition of

In the end the concept of a species is a man-made construct that tries to shoehorn the evolutionary diversity into pigeonholes. For example, your definition of breeding might work okayish for higher eukaryotes, but is unusable for prokaryotes. There people might be using some kind of definition based on the sequence identity of genomes. It's a bit like looking at a color gradient from black to white and then trying to classify each pixel to either black or white.