I would say appropriately enough this at this level it becomes an issue of semantics, and pragmatics, and specificity,
results like this AFAIK are real (have not been found irreproducible, or limited to e.g. Russian), but, what they demonstrate is best understood in the specific terms of their hypothesis and findings—the "risk" or possible alternative being, to extrapolate the "pernicious" lay belief into some strongish Sapir-Whorf "your reality/my reality."
I.e. measurably faster performance on discrimination tasks, does not map broadly to "perceives differently." Your phrasing was appropriately nuanced, "influences" is a reasonably lay summary...
...but "Japanese-as-first-language speakers perceive blue and green as differently as e.g. English speakers" is still definitely much more true than the alternative.
Personally I was sad to learn strong S-W didn't hold up. It is quite a captivating idea. So too this popular middle ground around color naming.
I find it quite interesting the way experiments like this are construed to try to tease apart linguistic, cognitive, and perceptual factors.