How to analyse Darlington pair?
The presence of Rc in the left schematic makes things complicated.  In a true darlington, the two collectors are directly tied together.  You can't just project Rc from the collector of the first transistor to the combined transistor on the right.  That would only be true if the two collectors were tied together, then Rc connected from there to the positive supply.

In this case, Rc only matters if it drops enough voltage to limit the current to the collector of the first transistor.  If it doesn't, then you can eliminate it.