What are the reasons for defining the autocovariance equation different from the covariance equation? Is it just to make quick simplifications for computing purposes assuming the number of "interesting" lags,k, is small?

According the following link (Autocorrelation Function | Real Statistics Using Excel) the definition of autocorrelation is given by

with the following observation

Notice the autocovariance divides by theentirecount of the time series observations and theentiremean of the time series observations. Whereas the regular definition of covariance is divided by the count ofeachseries and mean ofeachseries.

For large sets and small lags the consequences are minimized, but with few observations the effects are quite severe.