I thought heavy tail = fat tail, but some articles I read gave me a sense that they aren't.
One of them says: heavy tail means the distribution have infinite jth moment for some integer j. Additionally all the dfs in the pot-domain of attraction of a Pareto df are heavy-tailed.
If the density has a high central peak and long tails, then the kurtosis is typically large. A df with kurtosis larger than 3 is fat-tailed or leptokurtic.
I still don't have a concrete distinction between these two (heavy tail vs. fat tail). Any thoughts or pointers to relevant articles would be appreciated.
Best Answer
I would say that the usual definition in applied probability theory is that a right heavy tailed distribution is one with infinite moment generating function on $(0, \infty)$, that is, $X$ has right heavy tail if $$E(e^{tX}) = \infty, \quad t > 0.$$ This is in agreement with Wikipedia, which does mention other used definitions such as the one you have (some moment is infinite). There are also important subclasses such as the long-tailed distributions and the subexponential distributions. The standard example of a heavy-tailed distribution, according to the definition above, with all moments finite is the log-normal distribution.
It may very well be that some authors use fat tailed and heavy tailed interchangeably, and others distinguish between fat tailed and heavy tailed. I would say that fat tailed can be used more vaguely to indicate fatter than normal tails and is sometimes used in the sense of leptokurtic (positive kurtosis) as you indicate. One example of such a distribution, which is not heavy tailed according to the definition above, is the logistic distribution. However, this is not in agreement with e.g. Wikipedia, which is much more restrictive and requires that the (right) tail has a power law decay. The Wikipedia article also suggests that fat tail and heavy tail are equivalent concepts, even though power law decay is much stronger than the definition of heavy tails given above.
To avoid confusions, I would recommend to use the definition of a (right) heavy tail above and forget about fat tails whatever that is. The primary reason behind the definition above is that in the analysis of rare events there is a qualitative difference between distributions with finite moment generating function on a positive interval and those with infinite moment generating function on $(0, \infty)$.