You’re thinking of a concept called entropy, which is mot exactly the same as energy.
When you drop the ice cube in, you are disrupting a stable system of room temperature water, which increases the entropy of the system. Entropy in this case is referring to the fact that eventually all the energy will be evenly distributed in the glass of water, and the system is not yet. When the water is all room temperature again and the ice is melted, you can call this the low entropy state.
It is true that moving energy out of the water in the freezer takes extra energy, and doing this creates more “disorder” in the ice cube as it now has a greater equalization potential compared to the room temperature water; this is the high entropy state. You may notice that the farther apart things are in temperature, the faster they equalize, and they slow down as they near the same temperature.
You could technically consider each ice cube a sort of entropy battery to “store cold” but really you’ve just traded some energy for thermal entropy
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