“Statistically significant” means “the results are hard to explain as random.” Scientists often forget that not everyone is mathematically sophisticated; science journalists often went into science journalism because they think science is cool but long division made then have panic attacks.

I don't think that is a good example. I am not saying you are entirely wrong but in your example of flipping coins there is nothing worth doing more research on. The authors say statistically significant indicating it needs more research. They have targeted cheap drugs to perform a function they know has tertiary effects on cancer and in two studies the similar outcomes occur at a rate that is significantly significant. And just to define it for you. Statistical significance helps quantify whether a result is likely due to chance or to some factor of interest,” says Redman.

That particular example would not be statistically significant though- precisely because there's a decent enough chance of it happening randomly

What’s your null hypothesis in this example? Statistically significant means the probability of getting a more extreme result when the null hypothesis is true is less than the chosen value of alpha (typical 0.05 is selected for alpha). In the case of this article, statistical significance means that if the two populations (treated and untreated) respond in the same way (I.e., no impact of the drugs) the probability of the observed differences in the sample (or more extreme - greater differential with few secondary cancers in the treatment group) is less than alpha. Sample size is also accounted for in the statistical test. All else being equal you need a larger difference between smaller sample size values to be able to conclude statistically significant difference.

Statistical significance means "That's odd. Maybe e should get some funding to start a bigger study because we might have found something"