I think the parent comment is assuming ADT is short for "algebraic data types" and not "abstract data types", since the former is the subject of the rest of the thread. You used both long forms and the acronym without saying which you meant.

As to the meat of the discussion, algebraic data types imply that you can apply algebraic operators to types. Since this is traditionally a purely compile-time feature you can imagine why the parent comment is incredulous that you could implement this without language support.

Your specific statement, "data types and types are not the same thing," is particularly confusing. You are making assumptions that aren't warranted, for example that "data types" encompass both a compile-time type check and an interface allowing abstract operations, while "types" are simply static type guarantees on values. In fact, both terms can refer to either.