probably it’s not a bad idea to consult additionally some more recent studies to ponder whether humans and neanderthals admixed.
In  the authors show by a comparision of mtDNA of early modern human and neanderthals that there is no evidence for interbreeding. The mtDNA analysed does cover some of the claimed neanderthale/cro-magnon hybrids. The article excludes any large genetic contribution by Neandertals to early modern humans, but does not rule out the possibility of a smaller contribution.
The next study , based on a more realistic model of range expansion, shows that under this assumptions the maximum interbreeding rate is less than 0.1 %. If confirmed this would mean that early modern humans and neanderthales practically did not interbreed, and therefore should be treated as different species.
They are comparing figures from samples of modern populations, not the human population from that era, as one of the scientists quoted states, and thus the assertion that such genetic studies indicate 2 species is flawed, at least according to the scientists in the links provided.
I assume you are referring to the links you provided and specifically to the following quote:
"The problem with the DNA research was the interpretation," Dr. Trinkaus said. "It's demonstrably wrong. All that they showed is that Neanderthal biology is outside the range of living humans, not modern Homo sapiens back then."
Please note that the article of Serre et al.  addresses the criticism of Trinkaus. They describe their method in their introduction:
Thus, we did not attempt to determine DNA sequences that are similar to present-day human mtDNA. Instead, we determined whether Neandertal-like mtDNA sequences were present or absent in well-preserved remains of Neandertals and of early modern humans.
But maybe the problem lies in the term "early modern humans". Please have a look at the following link , which lists the fossils used by Serre et al.
First a recent taxonomic study which supports the hypothesis that neanderthals and modern humans are distinct species . I quote the abstract:
The taxonomic status of Neanderthals lies at the center of the modern human origins debate. Proponents of the single-origin model often view this group as a distinct species with little or no contribution to the evolution of modern humans. Adherents to the regional continuity model consider Neanderthals a subspecies or population of Homo sapiens, which contributed significantly to the evolution of early modern Europeans. Paleontologists generally agree that fossil species should be equivalent to extant ones in the amount of their morphological variation. Recognition of fossil species therefore hinges on analogy to living species. A previous study by one of the authors and recent work by other researchers [Schillachi, M. A. & Froelich, J. W. (2001) Am. J. Phys. Anthropol. 115, 157-166] have supported specific status for Neanderthals based on analogy to chimpanzees and Sulawesi macaques, respectively. However, these taxa may not be the most appropriate models for Pleistocene humans. Here we test the hypothesis that Neanderthals represent a subspecies of H. sapiens by comparing the degree of their morphological differentiation from modern humans to that found within and between 12 species of extant primates. The model taxa comprised >1,000 specimens, including phylogenetic (modern humans and African apes) and ecological (eight papionin taxa) models for Pleistocene humans. Morphological distances between model taxon pairs were compared to the distances between Neanderthals and modern humans obtained by using a randomization technique. Results strongly support a specific distinction for Neanderthals
Then another Neanderthal mtDNA study, whose abstract states:
During the late Pleistocene, early anatomically modern humans coexisted in Europe with the anatomically archaic Neandertals for some thousand years. Under the recent variants of the multiregional model of human evolution, modern and archaic forms were different but related populations within a single evolving species, and both have contributed to the gene pool of current humans. Conversely, the Out-of-Africa model considers the transition between Neandertals and anatomically modern humans as the result of a demographic replacement, and hence it predicts a genetic discontinuity between them. Following the most stringent current standards for validation of ancient DNA sequences, we typed the mtDNA hypervariable region I of two anatomically modern Homo sapiens sapiens individuals of the Cro-Magnon type dated at about 23 and 25 thousand years ago. Here we show that the mtDNAs of these individuals fall well within the range of variation of today's humans, but differ sharply from the available sequences of the chronologically closer Neandertals. This discontinuity is difficult to reconcile with the hypothesis that both Neandertals and early anatomically modern humans contributed to the current European gene pool.
The third articles analyses Neanderthal mtDNA data from the iberian peninsula  concluding that “Iberian Neanderthals were not genetically distinct from those of other regions” and
An estimate of effective population size indicates that the genetic history of the Neandertals was not shaped by an extreme population bottleneck associated with the glacial maximum of 130,000 years ago. A high level of polymorphism at sequence position 16258 reflects deeply rooted mtDNA lineages, with the time to the most recent common ancestor at ca. 250,000 years ago. This coincides with the full emergence of the "classical" Neandertal morphology and fits chronologically with a proposed speciation event of Homo neanderthalensis.