The "peakedness" definition of kurtosis is by now officially of historical note only and not useful as a descriptor.

Also, the fact that kurtosis is sensitive to outliers *is* the point of kurtosis. It measures outliers only, and nothing about the peak.

Westfall, P.H. (2014). Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195.