Pseudaelurus may be the most important cat in the entire family line that almost no one has ever heard of. Aside from ending the North American "cat-gap," Pseudaelurus is an important genus because these cats represent the last common ancestors of a diverse array of cats and "near cats." Saber-tooth cats, as a general term, technically fall out of the family line to cats before true felids. As an evolutionary grade, a group of taxa united by shared morphology, Pseudaelurus contains both felids and the true saber-tooth cats (Machairodontinae). This is why Pseudalurus is referred to as the last common ancestor to both saber-tooths and felids; the term "saber-tooth cat" can be extremely confusing because of the true and false labels in addition to the phylogenetic maze of carnivorans in which they settle out. For more information, I encourage everyone to read these articles on false saber tooths at ThoughtCo and Prehistoric Wildlife.
The genus Pseudaelurus has been separated and lumped a number of times over the years. However, the most recent phylogenetic studies (Werdelin et al. 2010 and Piras et al. 2013) have split Pseudalurus over three lineages definitively (unless someone comes along in the future to lump them again). The new genera include Hyperailurictis, Styriofelis, and Miopanther. These each represent a distinct lineage leading to the extinct lineage of American Hyperailurictis felids and the Styriofelis/Miopanther group (including both the extinct lineage of European Styriofelis felids and the extant Felinae which includes domesticated and wild cats). The third lineage retained the name Pseudaelurus and led to the extinct Machairodontinae, the true saber-tooth cats.