BJT Saturation Characteristic
I understand a saturation for an BJT is when the B-E voltage doesnt change the collector current anymore.


In the graph you presented, the x-axis is \$V_{ce}\$, not \$V_{be}\$. Changes in \$V_{be}\$ are represented by jumping from one parameterized curve to another. 

You can see in the saturation reason, the parametric curves overlap, meaning changing \$V_{be}\$ doesn't result in a change in collector current, just as stated in your proposed definition.