Does this procedure for extracting DNA from strawberries work?
The extraction protocol described in the first article is a standard DNA extraction protocol. I've used variations on this method in the lab (with some refinements) on other types of cells at least hundreds, if not thousands, of times. The "strawberry" experiment is very popular for classroom demonstrations, especially at the middle-school and high-school levels, because it's visually interesting and uses commonly available reagents. While these reagents aren't laboratory-grade, they're more than sufficient for a basic extraction of DNA.

I can't think of a scientific basis for why the author in the second article would claim that the extracted substance is "little more than pectin". Since he didn't provide data, or a reference to any data, it's hard to know what he's basing his conclusion on.

It's definitely possible that the DNA is pulling down a certain amount of cellular proteins during the alcohol precipitation step (the final step). This is why in a lab setting you typically perform a few more steps to "clean up" the DNA and remove any co-precipitating proteins. This would be essential if you plan to do further analysis of the DNA, because protein contaminants can interfere with subsequent analyses, making the data unreliable. One exception would be if you're studying protein-DNA interactions, but that's a separate type of experiment.

I'm assuming this experiment will be done for/by beginner level students, as opposed to a college or graduate level class? If so, then for the purposes of a classroom demonstration, those extra purification steps aren't warranted. It's time-consuming, and it would be very boring to watch... As is the case with most labwork. :-) Even if you perform those additional steps, the purified DNA is often precipitated with alcohol again, and it's visually similar to what you saw in the first alcohol precipitation, just a bit less opaque. Probably not something most students would notice or find interesting.

Either way, it's entirely acurate to say that the strawberry experiment is extracting DNA. You could always explain that the DNA might have cellular proteins binding (or "sticking") to it, and that a few other steps would have to be performed if you wanted to analyze the DNA in a lab setting, because the proteins would interfere with DNA analysis. This would also give beginner-level students an introduction to the fact that DNA and proteins interact with each other. 

Hope that helps!