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

Autocovariance vs covariance definition consequences-capture.png
with the following observation
Autocovariance vs covariance definition consequences-capture1.png

Notice the autocovariance divides by the entire count of the time series observations and the entire mean of the time series observations. Whereas the regular definition of covariance is divided by the count of each series and mean of each series.

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