In the old days you could by big blocks of ice and store them in your ice box to keep stuff cool. This later became a refrigerator. The very first snow cone maker was a crank that pressed a block of ice against some rotating blades to create shaved ice. The shaved ice was then drizzled with a flavored syrup. (Snow cone machines now are made to be safer and with cheap plastic, but the ice is not as thin and fine IMO.)
Ice cream does not actually contain ice and is just referring to the fact that it is chilled. Ice cream is more akin to a frozen custard (heavy cream, milk, sugar, and egg yokes) plus some sort of flavoring.