AFTER the contents of the fridge reach the exact temperature of the fridge itself (its setting), in the steady state the fridge contents have NO direct impact on fridge energy consumption. The fridge's system only runs to pump out heat that has come in from the outside. That all depends on the inside and outside temperatures, the effectiveness of the insulation, the door seals, etc. and has NO relationship to the contents inside the fridge. Sure, as heat flows in from outside, if the contents have a higher heat capacity their temperature will rise more slowly. But once the temperature is reached that triggers the compressor to come on, that same heat capacity means that the contents will take longer to release their heat and return to a lower temperature. So half-full vs. empty may have an impact on the length of running and not running, but overall the same heat flow (and hence same compressor hours of operation) per day will not change.
Potential confusion arises because people think not about ultimate steady state, but about the short-term period while a new equilibrium is being established. Your Case #1 - empty fridge - is the baseline that will establish how much compressor running is required to keep up with heat flow into the fridge from outside. Then Case #2 is half full of "cold" cans of soda/beer. If "cold" means EXACTLY the same temperature as the empty fridge, then there is no effect. But if the cans a just a little warmer, then the compressor will run more for a while to cool them down and establish the target state. From there on, we are at the steady state and fall into my first statements. Likewise if the cans are actually a little too cold when put in, the compressor will run less for a while until the heat flowing in from outside warms the cans up to the fridge setpoint, and then we've reached the equilibrium steady state. Case #3 is just an extension of Case #2b (cans are colder at first) but with the added factor that they are so much colder that a change of physical state is involved. It will take a lot more heat influx from outside to melt the cans' contents before they can warm up to the fridge temperature, so reaching the steady state will take much longer. But it still ends up at the state where the contents have no net effect; it just takes longer to get there. In the meantime we may not have enough patience to wait until the steady state is reached, what with cold beer calling out to us. So we may never manage to complete the experiment!