Typical of SUNRAY to accuse me of being wrong when I am not. Your first link shows a 14 element log periodic aerial wrongly described as a 28 element because someone has counted the elements twice. Your second link shows, as you say, an 11 element. I cannot see how any one would come up with 48 even counting the directors 4 times and the reflector 6 times.
The TV aerial industry as a whole do not count the elements several times, it is only some of the less reputable manufacturers, and electrical suppliers who know nothing about aerials that do this.
I cannot comment on the aerial behind your shed as the picture is not good enough.
No. the log periodic aerial linked to has 28 independant elements. I'm not going to insult you by describing how a log periodic is constructed. opposing halves of what behaves like a dipole are not one element, they are 2 elements, always have been described as 2 elements by the designer and I see no reason to redefine it.
Regardless of your opinion, if a manufacturer describes it as 48 elements then that is what one would have to order when purchasing. A member of the public wanting an aerial will not have a reason to order it as anything different to the manufacturers description.
Over the years I've purchased many
radio mic TV aerials from Alltrade (which incidently I usually cut down to about 4 elements; reflector, dipole & 2 directors) and, by inference, you describe them and the likes of Labgear & Antiference (and for that matter some Italian products) less reputable. So be it.
As to the aerial behind my shed, I warned the flash didn't penetrate far but you can see about a 40% of the boom and a dozen of the 29 directors. For clarification that is 29 pieces of aluminium ~6" long mounted across the boom, just the way you describe them. On this occasion you would call it 31 elements.