This is an excellent answer but there's one important mistake in it that needs fixing. It occurs twice.
1. "or, in more normal usage, unions and variants". A union is not a product type, it's an unsafe sort of sum type. (The safe sort is sometimes called a "tagged union".) And while "variant" means exactly what you want it to, it's a Microsoft-ism that not everyone will be familiar with.
So: in C-speak, product and sum types are structs and (tagged) unions. In Microsoft-speak, product and sum types are structures and variants.
2. "and 3D points a standard union ADT": again, it's not a union but a struct.
A few other remarks that may help people unfamiliar with this sort of syntax make sense of what you wrote:
3. That notation "Double -> Double -> Double -> Point" can be read as "a function taking a Double, and another Double, and another Double, and returning a Point". The weird notation is because actually it parses as Double -> (Double -> (Double -> Point)), and means it's a function taking a Double and returning a function taking a Double and returning a function taking a Double and returning a Point! The word to look up if that doesn't make sense but you'd like it to is "currying".
4. Relatedly, there is no distinction between a value of type T and a function of no arguments returning type T. That's why tel wrote "Zero : Nat" without any arrow anywhere on the right-hand side. You can (and maybe should) think of this as a function from (nothing) to Nat.
5. (This is implicit in what tel wrote but may be a stumbling block for the newcomer.) When you see something like "data Point = Point Double Double Double" it may not be obvious that the four things on the right-hand side are not at all on an equal footing. The first one (Point) is the name of a constructor. The others (Double Double Double) are the types of its arguments.
Ah, thanks! I tend to feel a bit hazy around "struct"/"union"/"tagged union"/"variant" as I feel the definitions are always language dependent.
On the other hand, product and sum (coproduct) are very well-defined terms which exist far outside any particular language. There's still haziness around choice of category (and, in particular, call-by-value versus call-by-name) but they're an order of magnitude more well-defined. I'd love it if the ADT concept became more widespread and could be easily used as a (low-level) description of more sophisticated typing like struct/variant.
To my ear "union" and "struct" are heavily overloaded since there was never a strict definition of what they should be. Product and Sum (Coproduct) are very well-defined (although, in practice, choice of category leads to some actual haziness especially around CBV/CBN).
In any case, I really appreciate the added detail about how better to use those terms and I'd love if the ideas of sums and products caught on and became a more stable point of communication.
1. "or, in more normal usage, unions and variants". A union is not a product type, it's an unsafe sort of sum type. (The safe sort is sometimes called a "tagged union".) And while "variant" means exactly what you want it to, it's a Microsoft-ism that not everyone will be familiar with.
So: in C-speak, product and sum types are structs and (tagged) unions. In Microsoft-speak, product and sum types are structures and variants.
2. "and 3D points a standard union ADT": again, it's not a union but a struct.
A few other remarks that may help people unfamiliar with this sort of syntax make sense of what you wrote:
3. That notation "Double -> Double -> Double -> Point" can be read as "a function taking a Double, and another Double, and another Double, and returning a Point". The weird notation is because actually it parses as Double -> (Double -> (Double -> Point)), and means it's a function taking a Double and returning a function taking a Double and returning a function taking a Double and returning a Point! The word to look up if that doesn't make sense but you'd like it to is "currying".
4. Relatedly, there is no distinction between a value of type T and a function of no arguments returning type T. That's why tel wrote "Zero : Nat" without any arrow anywhere on the right-hand side. You can (and maybe should) think of this as a function from (nothing) to Nat.
5. (This is implicit in what tel wrote but may be a stumbling block for the newcomer.) When you see something like "data Point = Point Double Double Double" it may not be obvious that the four things on the right-hand side are not at all on an equal footing. The first one (Point) is the name of a constructor. The others (Double Double Double) are the types of its arguments.