A Wikipedia scan by a layman (me) seems to say that is something that happened after their divergence from a common ancestor.
The dog’s similarity to the grey wolf is the result of substantial dog-into-wolf gene flow,[3] with the modern grey wolf being the dog’s nearest living relative.
The dog is a wolf-like canid.[7][8][9] The genetic divergence between the dog’s ancestor and modern wolves occurred between 20,000 and 40,000 years ago, just before or during the Last Glacial Maximum[2][1] (20,000–27,000 years ago). This timespan represents the upper time-limit for the commencement of domestication because it is the time of divergence but not the time of domestication, which occurred later.
Nope, genetically wolves and dogs are basically identical. Dogs are simply hundreds of generations of domesticated wolves.
A Wikipedia scan by a layman (me) seems to say that is something that happened after their divergence from a common ancestor.
You are correct. The ancestor of the modern dog is a now-extinct gray wolf different from the ones around today.