You're right that we would have just one species of bacteria. We ought to convince ourselves that we do have more than one species, though, and the authors of the article are hard to convince. They use the idea that the incredible diversity and rapid evolution that you mention aren't enough. This indeed relates to the term "evolutionary species."

The authors define "evolutionary species" as a population that evolves independently from others, meaning that genotypes can't cross the line between two evolutionary species. In the term "evolutionary species," I'd put the emphasis on the word species. Yes, all species are evolutionary, but not all evolutionary entities are species.

Why would anyone doubt that asexual organisms can split into evolutionary species? The article says that sexually reproducing organisms can diversify but also experience cohesion within groups, and "asexuals might not diversify into distinct species, because there is no interbreeding to maintain cohesive units above the level of the individual." Distinct is the key word in that sentence. Now, if asexuals have no source of cohesion (including homologous recombination), then they don't have a good way to form truly independent subpopulations, and they can diversify to the extreme, but yes, a group of (true) asexuals would remain always just one big evolutionary species. These authors and others point out that bacteria do form distinct populations (across which genotypes presumably can't cross), but those results are less convincing because the bacteria in question can all undergo recombination, if not sex proper, and that could induce cohesion. The problem of "one big species" still exists for totally asexual organisms like the rotifers in this study.

Figure 2 in the article http://dx.doi.org/10.1371/journal.pbio.0050087 shows all of these ideas about cohesion, species, populations, and so forth relatively elegantly.

In the end, this study shows that the characteristics of some truly asexual critters fit the model of cohesive, independent clusters better than not, and they also fit a model of selection based on more than just geographic isolation, for example. How do they cohere? Probably by ending up in niches, the authors say. It has always made sense that true asexuals could do this, but it sounds like we had little evidence about whether they did in fact, before this paper.

Turning to the Times and Slashdot articles, they are both right to point out that it's interesting that organisms can replicate themselves asexually for so long and still compete with sexual reproducers, but they don't emphasize why that's interesting. The comments are right to point out that both sexual and asexual reproduction can generate diversity, so that's not why asexuals are interesting. They're interesting because they lack the usual source of cohesion and independence. In short, the question was "can organisms diversify and cohere into real species or species-like units without any combining of their genes?" The answer is "it looks like it." As an aside, I have to apply a lot of effort to feel good when all tertiary sources seem to miss subtle points like this all the time.