Normalizing inputs

I’m a little bit confused about the terminology in the Normalizing inputs lecture. To the best of my knowledge from a statistical point of view, this is the definition of Standardization:
assigning the z value of each data point instead of the original value. or dividing the distance of each datapoint with mean, by standard deviation.
and this is the definition of normalization:
rescaling the values into a range of [0,1].
Although I think normalization can have a variety of meanings.
So wasn’t Standardizing inputs a better terminology?
and does anyone knows a universal definition of normalizing?
Thank you in advance for your comments.

As I understand normalization is a broader term for various methods aimed to adjust values to a relatively common scale. In that sense, there are various types of normalization: min-max, mean, unit norm, etc. and standardization can be seen as one of them.