I have no doubt there are regional variants on what, exactly, constitutes a 'mixed drink' as distinguished from a 'highball' or a 'cocktail'. My father in law, who worked for 40 years in a distillery, would have defined them as follows (and this is the usage in this part of the country):

'Mixed drink' and 'highball' are the same thing -- a hard alcoholic beverage, usually whisky of some kind, mixed with some carbonated beverage (usually), possibly plain water. Such as: blended whiskey (or rye) and ginger ale (the classic highball); scotch and soda, scotch and water, bourbon and branch, bourbon and ginger ale (classier than rye or rotgut), gin & tonic, rum & Coke, whiskey & Coke (ugh), to name the most popular. Vodka & OJ (screwdriver) might fit in this category.

A cocktail is a combination of liquor of some sort and (usually) another liquor (or 2 or more) or wine and (maybe) a smallish amount of some juice, soda, etc; e.g., the Martini (when made with vermouth, not just a glass of gin), the Gibson, the Old Fashioned, the Manhattan, etc.

A highball (or 'mixed drink') is properly served with ice in a highball glass, which is a tall cylindrical glass like an ordinary water glass, but holding about 6 oz. usually. A cocktail is served, either on the rocks or straight up in a smaller glass, which may be specific to the drink, like a Martini glass, which used to be unique to that drink but is now being used for other drinks. The other cocktails are usually served in what used to be called an Old Fashioned Glass, a squat tumbler holding maybe 4 oz.

The glass size demonstrates the difference between a cocktail and a highball -- the cocktail has more liquor proportionally; the highball is usually a pony (oz. and a half) and 5 or 6 oz. of mixer -- much less potent. You can drink a lot more highballs than cocktails in an evening.