Settling the issue of kurtosis once and for all

OK every time I pass by kurtosis I get confused, so I'm just going to write this all down here and hope the CFA gods here bail me out.



KURTOSIS refers to the peakedness of a distribution. So how 'sharp' a distribution graph is. OK so far.
LEPTOKURTIC (blue line) means that the graph has higher-than-normal peaks, and fatter tails than normal. Kurtosis is > 3 in this case.
PLATYKURTIC (pink line) means that the graph is flatter-than-normal, and thinner tails than normal. Kurtosis is < 3 in this case.
A normal distribution (red line) has kurtosis = 3.

Now my main source of confusion is the tails (the underlined parts). The books say that leptokurtic = fat tails, and platykurtic = thin tails, but how can you have fat tails with a higher peak? If most of the distribution is concentrated around the mean (high peak) then there will be little distribution around the extremes (tails), which means there will be 'thin' tails, wouldn't it?

Plus from the chart you can clearly see the blue line (leptokurtic) has thinner tails than the others.

What the hell? Really hoping someone can help me out because every time I see a question that requires me to know about tails I just cry.