If you look at research on carcinogens, it is frequent to see something like "it is statistically significant that eating chemical X increases your risk of cancer" but when you read on, the magnitude of the effect is "increases risk by .0023%, for a cancer that already has a .6% incidence in the population".
My point is that people who are not statisticians (or people who use statistics), tend to be concerned with the magnitude of an effect, and frequently misconstrue the statistical significance for that magnitude.
In fact, failures to reproduce "landmark" experiments have low significance (statistics word) because they find a result that is highly probable if the null hypothesis is true. But failures to reproduce have high significance (colloquial English word) because they indicate that widely-held beliefs are wrong. So in some of the most important cases, the statistics and colloquial English meanings of the words are exactly the opposite.
I don't know if there's a good solution to this, though--it's hard to change organically-emergent trends in language even if we did come up with a better term.
But these concepts aren't clear to the people using the data. I've seen people use statistical tests done in batch without correcting for multiple testing (one of my regrettable tasks), and flat out ignore any differences that weren't statistically significant. They thought it meant "no difference."
With population-level data (e.g., Census counts or hospital records), I sometimes wonder if statistical significance has any value. It's a tool to make sure we don't accidentally claim a difference where there isn't one. But I can guarantee you that nothing worth measuring is exactly the same for men, women, whites, blacks, young, old, whatever. It'd be amazing if the difference was 0. So statistical significance ends up being an obfuscated way to state the sample size.
The problem is that we are imbuing the words "statistical significance" with a whole bunch of math. This would be fine, except for the inconvenient fact that people also want to use the word "significance" as it is defined in English.
It is not only possible, but likely that people will be producing results that are insignificant but statistically significant. I mean seriously, if I tell my boss that the results are statistically significant, how do I expect him to understand that the results might reasonably be insignificant? How do I expect anyone non-technical to take that sentence seriously? You lose all credibility pretty quickly when people start saying that the result is significant except it doesn't matter.
This is even more stupid than calling complex numbers "imaginary" and "complex". Those names are just arbitrary. Statistical significance is overloading words that are likely to come up with meanings other than what they are expected to mean.
Anyway, this is the problem that things like this article are going to. People are treating significance values as though they are significant. They aren't significant, they are statistically significant. It means something different to significance.
For example; if I run an experiment 20 times and get a p value consistent with a 5% chance in one experiment, it is statistically significant but the result is obviously not significant in the English-language meaning of the word.