I think your last paragraph is correct, but the statement in the article is referring to the specific 2D 2-body example given, and its original phrasing is also correct. Translation, rotation, and time-shifting are transformations (matrices), not quantities. Horizontal, vertical, and angular (2D) momentum are scalars. The article is saying that if you take the action potential given in the example, there exist scalar quantities (which we call horizontal momentum, vertical momentum, and angular momentum) that remain constant regardless of any horizontal, vertical, or rotational transformation of the coordinate system used to measure the 2-body problem.
