The confusion is because that site is a British one, I can see how it makes little sense to everyone on here (and it could be worded better), but as we don't really have 7ft American tables over here (they are almost always 9ft, occasionally 8ft), a 7ft table would automatically be assumed to be English pool, which does use 2" balls.
The reference to 1 7/8" balls is because our homes are tiny and so a lot of home tables are 6ft with 1 7/8" or 2" balls.
That is interesting. And it makes sense to keep things proportional. In the US about the only time you see smaller balls is on one of those tiny desktop pool tables or on a small child size table.