If you mean the frozen juice concentrate sold in cans that you reconstitute four-to-one with water, then yes, once it's reconstituted we drink it and call it "juice." If you mean powdered stuff, like Kool-Aid...well, I think most people call it Kool-Aid, although the manufacturers probably dislike this
genericide. (There's our word for the day!) Crystal Light is a similar product without sugar for dieters; I've heard people actually call that "Crystal Light." ("You want a glass of Crystal Light?") I'm not sure I'm familiar with a syrup product like that, but I don't think we'd call it juice, probably "orange drink" (if it was orange) or "fruit drink" or whatever.
And no matter what the FDA calls it, nobody except the manufacturers calls anything a "juice cocktail." Some common forms of "juice cocktail," for example cranberry, are referred to as "juice." If you ask for a vodka and cranberry juice in a bar, you're almost certainly going to get vodka plus Ocean Springs brand Cranberry Juice Cocktail. But if you ask for vodka and orange juice, you'll get the real thing, though likely from concentrate, not fresh-squieezed.
From
Wikipedia:
"fruit juice can only legally be used to describe a product
which is 100% fruit juice, as required by the Fruit Juices and Fruit
Nectars (England) Regulations
[1] and The Fruit Juices & Fruit Nectars (Scotland) Regulations 2003.
[2]
However, the term "juice drink" can be used to describe any drink which
includes juice, even if the juice content is 1% of the overall volume.
[3]
In the USA, fruit juice can only legally be used to describe
a product which is 100% fruit juice. A blend of fruit juice(s) with
other ingredients, such as high-fructose corn syrup, is called a juice cocktail or juice drink[4] According to the FDA,
the term "nectar" is generally accepted in the U.S. and in
international trade for a diluted juice to denote a beverage that
contains fruit juice or puree, water, and which may contain sweeteners."