What is the meaning of 'primary' and 'secondary' sympatric speciation in this paper?

Question

Sympatric Speciation in the Genomic Era. Both terms are used throughout the paper.

I'm not able to make sense of these terms in the contexts used.

I've also heard the terms 'primary' and 'secondary' gene flow elsewhere on the same topic - are they related?

From the abstract:

However, I argue that genomic studies based on contemporary populations may never be able to provide unequivocal evidence of true primary sympatric speciation, and there is a need to incorporate palaeogenomic studies in to this field. This inability to robustly distinguish cases of primary and secondary divergence-with-gene-flow may be inconsequential, as both are useful for understanding the role of large effect barrier loci in the progression from localised genic isolation to genome-wide reproductive isolation. I argue that they can be of equivalent interest due to shared underlying mechanisms driving divergence and potentially leaving similar coalescent patterns.

From the introduction:

Lastly, I consider whether primary and secondary sympatric speciation represent a mechanistic dichotomy, I suggest that primary and secondary contact can leave a similar genomic signature, when speciation is driven by tightly clustered or large effect loci. Arguably, the advent of affordable population genomic studies should place less focus on whether study systems result from primary or secondary contact and instead focus on the mechanistic aspects of the genomic architecture and making progress in identifying the conditions and processes under which natural and sexual selection can drive speciation, without extrinsic barriers to gene flow.

Answer 1

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 "secondary sympatric speciation" or "speciation with secondary gene flow", "...with secondary contact" 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.