The ice in the drum will melt faster if the water is not draining out.
This is because any air space between the drum and the ice slug will be a good insulator.
As mentioned in a comment all the water will remain at 0 °C until all the ice is melted. Heat from the outside will cause convection currents in the water and transfer heat faster to the slug and melt more. Keeping the water in will allow for a maximum of drum surface in direct contact with the internal cold water and maximise the external heat transfer.
There is an added bonus if the drum is full, the ice slug will soon float free and rise up to touch the lid and gain further external surface area to assist with melting at the lower bulk ice temperature witch will have a slightly better thermal gradient and so be faster.
If the air is cold but the floor is hot then heat is mostly transferred through the bottom then it will be faster to drain the water.
In a cooler you want the maximum of cold thermal mass but the minimum of heat transfer. As the sides are an insulator the thermal mass is an advantage. If the sides were not an insulator like a tub or a drum then draining the water away from the warm tub walls is advisable but will only help while the ice has not melted to conform to the tub shape. The problem is that after the ice has melted you no longer have the small benefit of the cold water (that you drained). This means that you need to insulate to slow the heat flow and gain the benefits of the ice and the water. Using a foam yoga mat inside or outside the tub will improve things a lot in conjunction with some wood under it.
Keep in mind that melting ice is MUCH better at keeping drinks at 0 °C than water at 0 °C. Also ice water slurry is ideal for fast cooling of a warm beverage thrown into a tub or cooler as the water absorbs the heat faster due to better surface contact.