t-distribution peakedness and kurtosis

From my understanding (correct me if I am wrong), as degrees of freedom increases, the tails of the t-distribution become less fat. As a result, their peakedness increases. This relationship between fat tails and peakedness seems to me, to be contradictory to how leptokurtic is defined. leptokurtic is a distribution that is more peaked than normal and has fatter tails. If the peakedness increases in a t-distribution, how do its tails become less fat vs. a leptokurtic distribution that is more peaked and has fat tails?

Not too sure, I just know as degrees of freedom increase the distribution raises and reaches a peak limit of a normal distribution. And Lepto is more peaked than normal and Platy is less peaked, running out of room in my brain to start worrying about fat tails

Lepto has fat tails.

they are different curves I think but they do contradict, I posted something here about a week ago

for T distribution, the larger the sample, the more degress of freedom, the ditribution approcach normal distribution. therefore, the tail becomes less fatter and more data lie more closer to the mean and somewaht symetrical. remember now, when they say less fatter tails, they talking about ditribution with higher degrees of freedom versus one with less degree of freedoms. However, when you have an excexx kurtosis, you have bunch data lies away from the mean and a lot of data lie closer to the mean, so the data that lie way far far from the mean cause the leptokurtic to have a fatter tails.