I don't think you should be insulted for making this claim, but there are a lot of trolls out there that make it. So it is a bit like getting annoyed by a bunch of trolls running around claiming the sky is green and then running into someone who really sees the sky as green.
I've included a link to a demo that to me shows clearly the difference in frame rate.
That makes me think that it's a training & focus thing. Much like with MP3s, if you concentrate on the picture quality, you can see the difference. If you're just watching for the content, you can't. Most casual web viewers are looking for the content.
A better demo would probably be something like a fight scene. Close ups of fast moving objects very obviously show the difference (a classic mistake of bad cinematographers, which they often try to mask with darkness).
The human visual system is fascinatingly complex so something like this isn't too surprising.
High end digital televisions and cinema are having a hard time jumping to 60 fps which for many reasons would simplify things since the rest of the consumer electronics works at that, but traditional cinema is locked at 24 fps. There are various stories around you should be able to find that when cinema experiments with 60 fps (or even 48 fps), or compare televisions that can automatically upscale the framerate, consumers complain that it looks "cheap" because the quality matches what they see on YouTube or their home video camera or the evening news. People have been conditioned to accept 24 fps as high quality cinematic so they think it looks "cheap" when in reality they are getting more fidelity. (You should be able to find old reviews of The Hobbit and people comparing it to the evening news.)
To your original point though, in this case, people were able to distinguish the different frame rates. The difference necessarily register as frame rate to people, or paradoxically, necessarily even better.
The reason 60fps has become such a buzzword lately is because most laptop & phone screens refresh at 60Hz, and so you physically can't do better than that. At that frame rate, the electronics of the display become the bottleneck rather than your CPU/GPU power, and most engineers would rather say some engineer is the bottleneck rather than deal with the fuzziness of human perception.
Which is fine only for non-action scenes, and film also has motion blur to make movement look smoother.
Action scenes in games (especially without pseudo motion blur) look significantly better at higher than 30 fps.
24 fps was also chosen as "the minimum we can get away with", so I see no reason not to increase it when you don't have to pay for film stock.
Even the difference between 110 and 120 fps is clearly visible for most. And while sensitivity may vary: I doubt that anyone could not be able to instantly see the difference between 30 and 60. How was this tested? Sure that the frames were being delivered?
http://www.scholarpedia.org/article/User:Eugene_M._Izhikevic... shows some curves that show the effect of several of these factors.
In every day life I am constantly bothered by LED tail lights, and cheap video projectors. In both cases I clearly see the individual blinks in the case of the LEDs and the color cycle in the case of the projectors. I find in quite distracting.
Most people can't even see what I am talking about when I describe it and they go seeking it out.
Aside from that, one of the clearest videos I've seen the difference in is: https://www.youtube.com/watch?v=C-dOuBcxMlk - the commentary isn't very constructive in that video but I see a rather large difference between gameplay clips.