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Agreed, adding extra external insulation is always good - if the space isn't used for something else - but that was not the original question, which was about adding insulation inside the refrigerator cabinet.
SD - I don't understand your physics. I see it like this:
If you take an empty closed refrigerator, at ambient temperature as the starting point, and switch it on, the pump must cool down the cooling element and the air inside and then compensate for the heat leaking in.
Now repeat the experiment with a jug of water inside. The pump must now cool the water (but less air) and compensate for the heat leaking in. The cooling element is a constant in the comparison.
I think that water has a much higher specific heat than air, so more electricity will be needed to cool the water than to cool the empty refrigerator.
Once cooled, in both cases, the pump must compensate for the same heat leakage into the cabinet.
The value of extra insulation to reduce electricity demand, whether inside or outside the cabinet, is to be judged on its ability to lower the heat leakage. If an extra few centimetres could be added on the inside uniformly - so that the cooling element stays within the insulated volume - then the heat leakage is likely to be reduced.
Everything that is put into the cabinet has to be cooled, the more that goes in, the more that has to cooled. If less can get in, then less has to be cooled. So reducing the internal volume with insulation will reduce the need for cooling and electricity required.