Far simpler to explain, it’s a drink served with no ice in the glass.
Not the same as a ”Neat Pour” which is just pouring a measure of booze straight from the bottle into a glass. UP implies that a cocktail has been made. It’s cold, the dilution has been added and it is to be served without any ice.
It is usually presented in a stemmed glass because the stem allows you to hold the glass without warming the contents with your hands.
Whatever glass you use, it’s vital that the drink is PERFECTLY DILUTED and COLD ENOUGH before it is served. As once served, no more water will be added and it’s only going to get warmer while it sits in the glass.
Keeping glasses in your freezer for these drinks is another nice touch, and just helps to keep the drink ice-cold that little bit longer.