A while ago I asked a question asking what is field strength renormalization (What exactly is field strength renormalization?). I now have a better way of thinking about this, which is that it relates the one particle state of the free theory to an interacting theory. This renormalization changes the the probability of a one particle state being created after the field acts on the vacuum from 1 (in the free theory) to $1/\sqrt{Z}$ (in the interacting theory). We call $\sqrt{Z}$ the field strength renormalization. (As a side remark, this does mean that $Z \geq 1$ so that the probability $1/\sqrt{Z}$ is bounded above by 1? Why do we take the square root of $Z$?)
However I am still confused on what is the difference between field strength renormalization and wavefunction renormalization. Many sources, including answers in the question linked above, state that they are the same. Other sources say that the wavefunction renormalization is $Z$ whereas the field strength renormalization is $\sqrt{Z}$. Which one is correct? If they are the same, why do both terminologies exist but are used in different contexts? If they are different, what is the difference other than one is $\sqrt{Z}$ and the other is $Z$?
