This is a partial duplicate of [this Stack Exchange question][1] which unfortunately got no answer.

All schemes are Noetherian and of finite type, although they need not be normal.

With $Z \subset X$ a closed subscheme, consider the blow up $\pi: \operatorname{Bl}_X(Z) \rightarrow X$. $Z$ is called the center and the strict transform of a subscheme $W \subset X$ with $W \not\subset Z$ is then the Zariski closure of $\pi^{-1}(W \setminus Z)$.

In my research, I must analyze some certain blowups. In doing so, I need to talk about certain subschemes which I feel should have a standard name but I have not been able to find. They are as follows, with my (nonstandard) terminology.

1. The "proper center": The closed subscheme of $X$ where $\pi$ is not an isomorphism (i.e. where the subscheme $Z$ is not Cartier).
2. The "true center": The closed subscheme of the proper center where the fiber above each point has dimension greater than zero.
3. The "quasi-finite center": The complement of the true center inside the proper center.

Are there accepted names for these? Furthermore, I also need to calculate the strict transform not with respect to the center, but rather the proper center. Is there a name for this as well? 

Finally, I am also wondering if there is any expository reference where the strict transform is taken with respect to the proper center.

  [1]: https://math.stackexchange.com/questions/469128/terminology-for-blow-ups-in-algebraic-geometry