What are the parallel branches shorted out in one but not in the other case?
It's because in the 2nd case, you can "merge" \$R_1\$ and \$R_2\$ because they are in parallel. Then, going in the way of your logic, if \$R_1\$ is shorted, then the equivalent resistor to \$R_1\$ and \$R_2\$ is also shorted.

Current is lazy, it rather use a resistanceless path than another one containing resistor to go to a point to another. 


In the first case scenario, there is a path without any resistor to go to point "A" to point "B". But this is not the case for the point "C" and "D" due to \$R_3\$  and \$R_4\$








simulate this circuit &ndash; Schematic created using CircuitLab