I think when one uses these different plant lists it is important to recognize that these are constructed based on what is out there in the general and scientific literature and what the compilers of the specific list decide to accept as what is correct about a plant name - sometimes possibly ignoring literature stating something different. For example the decision to merge E. eyriesii into E. oxygona was first presented to most of the general cactus growing public by D. Hunt in the New Cactus Lexicon in 2006 - this is the ultimate source for ITIS to list E. eyriesii as a synonym of E. oxygona, ITIS appears to be the ultimate source for the Catalogue of Life in this case. So while CoL might have been updated more recently than the Plantlist, the basic info that has CoL declare E. eyriesii a synonym of E. oxygona has been out there since 2006. The fact that the Plantlist does not report it as such must mean that the compilers of that list do not deem Hunt's info (unfortunately I only have the illustrations volume of the NCL, not the text volume with the references, so cannot track down Hunt's reasoning/source) convincing enough to also declare those two species synonymous. The Plantlist quotes a different source of data that is also hard to track down, so it is unclear exactly what the different listings of these plant's names is based on, and so hard to evaluate what is correct. It illustrates how some of cactus genera still need a lot of work to sort out exactly what is what, DNA analyses will help and may have been the source for declaring these two species synonymous, but as said I could not track that down.
Interestingly, for me the two plants that I bought as eyriesii and oxygona are very different and seem to fit well to their respective descriptions as found in Anderson's 'The Cactus Family' (2001), E. oxygona has longer spines are more pink-purple-reddish blooms, while E. eyriesii has the short spines and white blooms. My short spined pink-white cactus came labeled as E. x 'Los Angeles' and looks a lot like the plants in the OP, while it looks similar to my version of E. eyriesii, overall body size and shape (and for me flower color) are different.
One suggestion I read somewhere was, that if these two plants are indeed examples of intraspecies variation, to keep the name 'eyriesii' as a variation or cultivar name so that the distinct short spined plant generally with white blooms can still be named as such in the plant trade (to keep the collectors happy).