Here are some of the terms I have seen.

1.  The *image of the exceptional set(/locus)* is probably the most explanatory.  The word center is also often used for the image of a single irreducible component of the exceptional locus so you might run into confusion if you call it the true center.  I've also seen this called the *discriminant*.

I don't think I've ever seen people give names to 2 and 3, but I've seen number 2. show up.

**EDIT:**  As pointed out in the comments, I was surprised to learn that the strict transform is taken with respect to $Z$ (or rather $X \setminus Z$) instead of depending simply on $\pi$.  In birational geometry the actual ideal you are blowing up doesn't seem to matter, people just care about the map.  In that setting, the strict transform is also often called the *birational transform*.  I've also seen the *proper transform*.