(1) The term standardization is often used as a generic term that could refer also to other methods of this kind such as standardization to 0-1 range, or to median 0, MAD 1.
(2) In many cases standardization is done in order to apply some method to the data that requires that the variation of the different variables is comparable. Almost all such methods are invariant against shifting the mean (or will be "equivariant", i.e., will adapt accordingly), such as for example computing Euclidean distances or interpreting regression coefficients, meaning that it doesn't matter whether you also subtract the mean or not.
(3) Standardization to mean zero, sd one has the advantage that values of the different variables can be interpreted in a unified manner as differences from the mean in sd units. Not subtracting the mean means that there are still meaningful mean differences between variables. I can imagine this to be useful in some cases, but frankly I have never come across such a situation, so I have always either also subtracted the mean, or done something entirely different, never just divided by the sd (although the "advantage" of standardizing to mean 0 mentioned above is also often irrelevant).