When did birds and mammals diverge?
The Most Recent Common Ancestor (MRCA) of birds and mammals was the first amniotes.

Fossil record

The latest known fossil of amniotes are 312 million years old (Bentou and Donoghue, 2007).

Genetic data

From EOL, using data from The Paleobiology Database, the first amniote is in between 314.6 and 323.2 million years old. Those results are a little bit older than the latest known fossil which very much makes sense.

The (apparent) contradiction in the sources you found


  This site seems to imply it was at least 250 milions of years ago.


They actually say 250 to 320 millions years. This broad estimates match the above studies and data.


  This scientific article mentions 600 milion years but the sentence is a little unclear.


It is actually a news feature published by a science writer (former neuroscientist according to her twitter account), not a researcher. She indeed made the claim (Even so, 600 million years of evolution and radically different brain architectures separate birds from humans.). There is no reference for the claim and based on above evidences, it appears at first to be a mistake. It is not impossible though that she considered the evolutionary time in both branches in her statement but that would be very unusual If two lineages diverged $t$ years ago, then they both evolved for $t$ years and it could arguably be said that they are separate by $2 t$ years of evolution. If this is what she meant, then the claim is correct but quite misleading.