They should be the same. Precise, universal definitions are essential to good science. We don't define a meter differently depending on whether we are observing a meter in Europe or in America, or whether we are talking about a meter in sedimentary layers or in a laboratory.
And, of course, we should be seeing more than speciation. There should be an observable rate of organisms moving into new genuses as well as new orders, classes, families, etc.
You see, evolution also requires not just speciation, but phylum, family, class, order, and genus creation as well.
So what should the definition of species within a genus be?
"They should be the same."
Then come up with a definition of species that can be used on both living organisms and fossils. If you claim it can be done, then do what noone else has been able to do.
Noone has even been able to come up with a definition of species that fits living organisms alone, that doesn't have contradictions.
"And, of course, we should be seeing more than speciation. There should be an observable rate of organisms moving into new genuses as well as new orders, classes, families, etc."
This would take far longer than even speciation. Ring species already show speciation to be a fact of life. Are you going to even address ring species?