Every cocktail with ice in the mixing glass or tin is doing two jobs at once: it is getting cold, and it is getting wetter. Both come from the same melting ice, so you cannot have one without the other. The only real question is how fast you get there and what else happens on the way.
Melting ice soaks up a surprising amount of heat. Turning a gram of ice into water takes about 80 calories, the same energy that would heat that gram of water from 0 to 80 °C. So when ice melts into your drink, the drink gets colder and the melt water goes into the glass. Bartenders call that water dilution. A well-made stirred drink ends up with roughly a quarter to a third of its final volume as water, and that is a feature: it softens the alcohol, opens up the flavour and takes the burn off.
Drag the slider and flip between the two methods. Stirring is a slow, gentle climb: around 30 seconds gets a spirit-only drink to about -2 °C with roughly a quarter of water added, and 90 seconds pushes it a little colder and to around 32 percent. Shaking gets to the same place in a fraction of the time. Published measurements put a hard shake at around 12 seconds before the drink reaches thermal equilibrium, at roughly -5 to -8 °C, and further shaking changes very little. One bar guide sums it up as 15 to 20 seconds of shaking matching 90 to 120 seconds of stirring.
Because speed is not the only thing shaking does. Hard shaking breaks the ice into chips and whips air into the liquid. That is wonderful when you want it and a nuisance when you do not.
Citrus juice, cream, egg white and anything else cloudy wants the air. It gives a lime-based Daiquiri a lighter, livelier texture and it is the only way to get cream or egg white to mix properly. Spirits and vermouth, though, cannot hold bubbles, so a shaken Martini goes cloudy and then slowly clears as the air escapes. Nothing is harmed, but you served something that looks unfinished. So the common rule is simple: if every ingredient is clear, stir; if there is juice, dairy or egg, shake.
Stir what is clear, shake what is cloudy.
Fill the mixing glass with plenty of cold, dry ice, add the liquid, and spin the bar spoon smoothly against the glass for 20 to 30 seconds. The goal is for the ice and liquid to move together, not for the ice to rattle. Taste a drop from the spoon: it should taste cold, smooth and clearly alcoholic, but not hot. If it still bites, stir a little longer. Strain into a chilled glass.
Fill the tin most of the way with ice, shake hard for around 10 to 15 seconds, and stop. After that you are just working your arm. Use fresh, dry ice if you can: wet ice from a freezer that has been opened a lot is already part melted and gives extra water on top. For a drink served over ice, shake a bit shorter, because the ice in the glass will keep adding water as it sits.
Less than you might think for the shaken or stirred result, as long as you adjust time. Smaller ice melts quicker, so it reaches the same temperature and dilution sooner; bigger cubes get there more slowly. What big, dense ice is really good for is in the glass: it keeps a drink cold without drowning it, which is why an Old Fashioned goes on one large cube.
Chill the glass first (a few minutes in the freezer, or fill it with ice and water while you mix), so the drink does not warm up the moment it is poured. And measure everything: dilution is a percentage of your starting volume, so a loose pour changes the balance before the ice does anything.
As always, enjoy it slowly. A well-diluted drink is meant to be sipped, and an alcohol-free version of any of these, such as tea or juice mixed the same way, gets cold and balanced by exactly the same rules.