It's nice that people are looking to understand the evolution of our breed because the more questions people ask, the clearer it will be.
Poodle in the years 1900 at a dog show in Paris (on the left, top). Compare to Mahler 1907(on the right, top), one can see the 2 breeds have nothing to do with each other, and haven't in 50ish years, even then in 1900.
Poodle has a long nose, look at the tail carriage, the one on the right has pretty straight angulations.
Barbet has a nice square muzzle, not a loooooooooong skinny body, tail carriage, of course depending on whether they are static, excited or whatever, ( and implantation) is different.
Last but not least, the coat texture, and curls are completely different.The Barbet coat having an oily texture to it.The poodle's being dry, frizzy and must be kept short (or corded?)A Barbet doesn't cord.
Here you have a Portuguese water dog and it is "easy" to then understand how, when wanting to re-incarnate a Barbet Moderne, poodles were crossed together and then was inadvertantly added some Portuguese Water dog blood.
They are of a similar type, but character and aptitude are different from that of the Griffon Barbet which was a hunter who loved water.
Some poodles can still hunt, but not many.