What is cold ice?
Ice melts at 32F so we know that ice cannot be warmer than 32F. But it can be colder. Many restaurants’ freezers are set at OF which means ice pulled from a Freezer will be at OF. So?
So, cold ice will absorb more heat out of your drink before it starts melting, and diluting, your drink. “Warm” Ice pulled from the bar basin where it’s been sitting, and melting, for a while will start diluting your drink immediately upon being placed in the shaker.
In a perfect situation the bar would use cold ice from a freezer in the shaker when preparing the martini (shaken or stirred). The ice would chill the gin and just as it reached 32F, and before it starts to melt, the barkeep would pour your Martini. Fully chilled and totally undiluted.