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And to add to Olin Lathrop's answer, case 1. might be called "multidrop" or "party line" or "dot-or" connection.  IIC bus as a good example of WHY we might use this case.

Case 2. allows what might be called "level shifting".  For example a digital system using a 3.3V power supply might be used to control a 5 V system.

An open-drain output of a 3.3 V digital system can control a 5 V input if the 3.3 V open drain output can withstand the 5V when it is in the high state.  If so, we might say it's "5 V tolerant".  If we connect that 5 V input with a resistor to its own 5 V supply, it will be in its high state.  This is called a "pull up resistor".

Then if we also connect that 5 V tolerant open drain output from the 3.3 V system to this same point, and connect the two systems' ground points, then the 3.3 V system can control the 5 V input.  The low state is near zero volts because the 3.3 V system pulls it low, the high state is near 5 V because the pull up resistor connects it to the 5 V of that system.

There are many examples of using this connection to control 5 V output devices on 3.3 V powered computer systems.