Here are some thoughts

1. I think this should be called the "true center", because it is where something interesting happens. Plus, if you blew up,say, the singular point of a nodal curve, then according to your definition the "true center" would be empty which doesn't sound right. As Karl suggests calling it the "true center" could be confusing, especially given your original feelings about this, so you could call it the non-Cartier center, but I think this is really the true center, so you might as well call it that.
2. This is probably the only one among the three that already has a name. I believe classically this locus was called the fundamental points of the map. You can find this terminology for example on page 156 in Mumford's Algebraic Geometry. Another reasonable name would be the exceptional center. I think that's rather self-explanatory.
3. I'd be quite surprised if this locus did indeed have a name. This should definitely not be called a center, since it is generally not closed. Actually I don't see a reason for this to have a name. In what situation would you need this as a set other than say that a point lies in it? Given the increased complexity induced by having to distinguish between the other loci, you could just refer to this as the difference of the above two sets. Or if you really want a name you could call these the non-fundamental points of the non-Cartier center or the non-exceptional locus of the true center, or other combinations of the above.
4. I also don't think that anyone used a different name for the strict transform with respect to the non-Cartier center, but it feels like this should be the true strict transform. You could also just call it the strict transform with respect to the non-Cartier center.

Here are some thoughts

1. I think this should be called the "true center", because it is where something interesting happens. Plus, if you blew up,say, the singular point of a nodal curve, then according to your definition the "true center" would be empty which doesn't sound right. As Karl suggests calling it the "true center" could be confusing, especially given your original feelings about this, so you could call it the non-Cartier center, but I think this is really the true center, so you might as well call it that.
2. This is probably the only one among the three that already has a name. I believe classically this locus was called the fundamental points of the map. You can find this terminology for example on page 156 in Mumford's Algebraic Geometry. I originally suggested that another reasonable name would be the exceptional center, but after rightndsd's comment I agree that it is not a great name as it suggests that it is the image of the exceptional locus. So, how about fundamental locus? Its preimage could be the prefundamental locus.
3. I'd be quite surprised if this locus did indeed have a name. This should definitely not be called a center, since it is generally not closed. Actually I don't see a reason for this to have a name. In what situation would you need this as a set other than say that a point lies in it? Given the increased complexity induced by having to distinguish between the other loci, you could just refer to this as the difference of the above two sets. Or if you really want a name you could call these the non-fundamental points of the non-Cartier center or the non-exceptional locus of the true center, or other combinations of the above.
   ADDED OK, how about the residual locus (of the true center)? Or if you adopt fundamental locus for the above, then this could be the non-fundamental locus...
4. I also don't think that anyone used a different name for the strict transform with respect to the non-Cartier center, but it feels like this should be the true strict transform. You could also just call it the strict transform with respect to the non-Cartier center.