Logic behind Zener regulator conditions
Think of it this way:

As the load resistance decreases, more current flows through the load, and therefore less current flows through the Zener. When the source voltage is lower, even less current flows through the Zener (as less current flows through the entire circuit). If the load current is very large, the current through the Zener drops below \$I_{ZK}\$, the knee current of the Zener and it stops behaving like a Zener (so we are no longer in the "proper operation" regime). This gives rise to the first point. Additionally, if the source voltage drops below \$V_{Z0}\$, then the Zener will definitely not be acting like a Zener anymore.

If the load resistance increases, the load current decreases and current through the Zener increases. When the load current is zero (no load) and the source voltage is at a maximum, that is when the most current flows through the Zener. In order to avoid breaking the Zener, the power dissipated by the Zener must be below its rated value, and this gives rise to the second point.