The top circuit is OK in principle. The bottom is not. You cannot use an NPN transistor as a high side driver in such a circuit.
However, the BC337 is not suitable for most motors, as it not designed for typical start/stall currents, which exceed the 800 mA absolute maximum collector current.
I have found in general it is more cost effective to buy a bridge driver then to design and build one. The upper part T1 and T4 may work as emitter followers but the output will be about 0.7V less then the base voltage which will help them get nice and hot driver voltage (A, B) and base resistor voltages (R1, R3) have not been included.
My goal is to understand certain concepts which is why I don't buy a bridge driver yet.
I didn't quite understand the last sentence. Were you explaining to me the reason for the transistor overheating in the second configuration?
I still don't understand why the transistor heats up more in the second configuration and why I can't use an NPN transistor as a high side driver in this circuit.
As any textbook on transistor circuit design will explain, the voltage on the transistor emitter can never be higher than (base voltage - 0.7V), which in this case increases the transistor power dissipation. So it heats up.
If you want to actually learn this material, randomly chosen web sites are usually not the best sources of information.