The word "normalization" can mean many different things, as this page and this page nicely illustrate. It doesn't necessarily mean making the transformed values "appear as though they have a Gaussian distribution," although it often does.
A Gaussian distribution should provide probabilities for arguments over the entire real number line, $(- \infty, \infty)$. Both transformations you cite in your question map non-negative grades into necessarily positive transformed values, so strictly they aren't producing normal distributions. These seem to be ad hoc attempts chosen by trial and error to look good when you plot histograms of the transformed grades, and perhaps also to meet a university's idea of what an average grade and the distribution of grades among students taking a course should be.
If you want to force the transformed grades to match a chosen or estimated Gaussian distribution, you could place the grades in order to determine their quantiles (percentiles), then use the inverse of the Gaussian distribution to get the values along the real line to which those quantiles would correspond. This is called the "rankit," which when plotted with the actual quantiles in a normal probability plot provides a nice graphical display of how well the original data match what would be expected from a Gaussian distribution. If the original data don't follow a Gaussian distribution, however, grades close to each other in the original scale could be quite far from each other in this type of transformed scale.
Power transforms like the Box-Cox transformation can be used to make the distribution of data points seem more like those expected of a Gaussian distribution while being less likely to lead to wide discrepancies of transformed values between close neighbors on the original scale.