What is the difference between normalization and re-scaling?

This site does not describe the nature of the tag. How does it differ from re-scaling? Many authors use the two terms interchangeably. I can not understand normalization's operational meaning.

Re-scaling means that you multiply your variable by a factor, i.e. $$x \to x/a$$. A normalization is a specific kind of re-scaling, where the factor $$a$$ is such that the values of $$x$$ become of order one. Its form depends on the context and what you are trying to do. Examples are $$a= \langle x \rangle$$ or $$a= \textrm{max}(x)$$. Perhaps the most common one is when you do $$x \to (x-\langle x \rangle)/\sigma$$, because if $$x$$ follows a Gaussian distribution, the re-scaled variable follows a normal distribution (average 0 and std 1). Note, this last transformation actually involves a re-scaling, but also a translation.
• "Your variable" means a random variable your aim to re-scale. $x$ is the random variable. $a$ is the factor. Feb 22 '21 at 1:30
• In which sense $x \to (x-\langle x \rangle)/\sigma$ is not a re-scaling (after a translation) and, more specifically, a normalization? Feb 23 '21 at 16:45