Kurtosis is really a measure of the outliers (tails) of the distribution. It is unfortunate that sources still persist with the incorrect “peakedness/flatness” description. It’s not really your fault — many people keep repeating this misinformation, and the error propagates.
As far as kurtosis and “fake news”, there are two points to be made. First, high kurtosis is an indication of an outlier (or outliers), and certainly oultiers need to be investigated in any context. They could be mistakes, or “fake news,” or something else. It’s hard to know without further investigation.
Low kurtosis, rather than being a measure of “flatness,” indicates a distribution that is less outlier-prone than is the normal distribution. As such, it could be an indicator that there is some trimming of unwanted results — data that are “too good to be true.” Some cases of academic fraud have been discovered in that the published data were “too good to be true.”
So I agree that either case, whether low or high kurtosis, bears further investigation; perhaps either is an indication of “fake news.” But it is because of outliers, or perhaps unwarranted lack thereof, that is the issue. Peakedness/flatness is not only irrelevant to the “fake news” angle, it is dead wrong as an interpretation of kurtosis.
BTW, the Medical dictionary pictures are misinformation too. All they show is distributions with different variance. But kurtosis is scale-free, so you should make all variances the same (e.g., 1.0), for a valid comparison. The Medical dictionary pictures themselves are “fake news”!