What is the meaning of 'primary' and 'secondary' sympatric speciation in this paper?
TLDR
Sympatric speciation and allopatric speciation with later migration into the same habitat were historically diffucult to distinguish without looking at palaeo-biological data. The paper argues that while palaeo-genetics has made this easier, it is still difficult to distinguish pure sympatric speciation (which it calls primary) and sympatric speciation with a geneflow from an geographically separated (allopatrically speciated?) subpopulation (which it terms &quot;secondary sympatric speciation&quot; or &quot;speciation with secondary gene flow&quot;, &quot;...with secondary contact&quot; etc.).
Speciation
Speciation is the divergence of one species (with one gene pool) into two different species (with different gene pools). It is obvious that this will happen if subpolulations are geographically separated and continue to adapt to their local conditions (allopatric speciation).
However, Mayr suggested (back in the 1940s) that there is another type of speciation that happens while the speciating populations share a habitat, and, consequently, while gene flow between these subpopulations is maintained until the speciation process is complete. This requires strong selection pressure towards two different ecological niches each with their associated adaptations.
Empirical examples have been discussed and called into question again. One cool and frequently discussed example is that of the apple maggot in North America that has developed from the hawthorn maggot after the introduction of apples in North America.
Debate
Unequivocal examples for pure sympatric speciation are rare, leading to some debate about the merits of the concept. Sympatric speciation models (following Maynard Smith's models from the 1960s) as well as the reasons for skepticism towards them are illustratively explained for instance in this article (Felsenstein 1980), page 133-135.
The debate continues, today with additional, newer findings from genetics, palaeo-genetics, etc. This is what the article in the OP (Foote, 2018) was about. The pdf of the article (Foote, 2018) without paywall can also be found here. While Foote suggests that what pure sympartic speciation and speciation with secondary gene flow may never be unequivocally distinguished (without additional palaeo-genetic information), others are more optimistic, e.g. Richards et al. 2019 (without paywall on biorxiv). Coincidentally, Richards et al. have a very nice illustration (their Figure 1) that illustrates the issue the OP was asking about.