I believe you're an reasonable, intelligent layman. Do you believe we're reasonable, intelligent people too? Are you willing to consider the possibility that the article did not make an "appalling oversight", the article merely phrased things confusingly?

You and I both have a good idea of what we think of as a two-dimensional surface, and as a three-dimensional space. In particular, we have an idea of how many real numbers uniquely identify any point in the surface or space (2 and 3, respectively), and we call these coordinates. Now comes the important part: we also have an idea of how to measure the length of any path in the surface or space, and as a derivative idea, the idea that the distance between any two points is the length of the shortest path between them. On a flat two-dimensional surface--among the English-speakers who converse often about such things, this is called a "plane"--it turns out that the distance between any two points is exactly the square root of the sum of the squares of the differences between the coordinates of each point (in other words, sqrt( (x1-x0)^2 + (y1-y0)^2 )). There is a similar relationship between distances between points and their coordinates in three-dimensional space, right? I'm sure you've seen all this before.

Now, there are two-dimensional surfaces for which that relationship is untrue, for example, the surface of a sphere or a surface that is "saddle-shaped" (like in the diagram, although they don't (and can't) show it extending infinitely). In these two examples, it turns out we can describe such surfaces, rather than as a set of points each uniquely determined by 2 coordinates, instead as a set of points uniquely determined by 3 coordinates and one constraint on what those coordinates can be, where the 3D distance has the relationship with coordinates that we're used to; for example, the surface of a sphere can both be thought of as being uniquely determined by longitude and latitude, and as being uniquely determined by x,y, and z coordinates, subject to the constraint that every point in the surface is the same 3D distance from some point in the 3D space. When described this way, things that would be non-obvious if I told you the relationship between distance and latitude and longitude, such as the fact that the 3D sphere is closed, become very obvious, because we have a good intuition for distances like that.

Now consider the following: let's say you're trying to figure out how the world works. An approach that's been quite successful is describing all the possible ways you can think of that are consistent with what we know, and then conducting experiments to try to rule each possibility out. One possibility you're describing involves the points being uniquely identified by 3 coordinates, but the distance between points doesn't have the usual square-root-of-the-sum-of-the-squares-of-the-differences relationship with the coordinates, and is instead just-so-slightly off. Let's say it is off in such a way that by describing the same set of points with 4 coordinates and 1 constraint (similar to how we described the sphere as 3 coordinates and 1 constraint), where the 4D distance between points uniquely described by those 4 coordinates obeys a relationship to the coordinates that is very similar to intuitive distances for the flat 2D surface and 3D space, so similar that almost all the same reasoning applies (continually increasing a coordinate, for example, will eventually continually increase distance in these cases, but not for longitude and latitude of a sphere).

Don't you think describing this as a "curved three-dimensional surface" is a sensible and accurate description?