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.

I'm not quite sure what you mean by "where the strict transform is taken with respect to the proper center". Isn't the strict transform of $Y \subseteq X$ defined as follows? Take the pre-image of $Y \cap \big( X \setminus (\text{proper center})\big)$ in your terminology, and then take the closure in $Bl_X(Z)$? I thought this was always how you do it? You mean people instead intersect $Y$ with $X \setminus Z$?

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.