The first two coordinate systems (1),(2) you listed are in CCF coordinates where zero is at the corner of the bounding volume as shown in this image:
Note that we don’t have a bregma location in CCF as described in this post
In (1), the SDK tutorial guides you to download gridded projection density data. It defaults to downloading data at 25 micron resolution. Hence each voxel of the [528, 320, 456] grid is 25 micron isotropic.
In (2), the “Location” in Brain Explorer 2 display CCF coordinates in millimeters (mm)
System (3) are the coordinates used in each case for stereotaxic injection. Below is a snippet from the whitepaper describing these coordinates:
Selection of Stereotaxic Coordinates
Atlas-derived stereotaxic coordinates were chosen for each target area based on The Mouse Brain in Stereotaxic Coordinates (Paxinos and Franklin, 2001). For the majority of target sites, the anterior/posterior (AP) coordinates were referenced from Bregma, the medial/lateral (ML) coordinates were distance from midline at Bregma, and dorsal/ventral (DV) coordinates were measured from the pial surface of the brain. For several of the most caudal medullary nuclei ( e.g. , gracile nucleus and spinal nucleus of the trigeminal, caudal part), the calamus (at the floor of the 4th ventricle) was used as a registration point instead of Bregma. For many cortical areas, injections were made at two depths to label neurons throughout all six cortical layers and/or at an angle to infect neurons along the same cortical column. For laterally located cortical areas ( e.g. orbital area, medial part; prelimbic area; agranular insular area), the injection was made at two adjacent ML coordinates as the pipette angle required for injection along the cortical column was nearly 90° and technically infeasible.
We cannot convert between (1) and (3) because (3) is respect to a skull base landmark and that information is lost when the brain is dissected and processed.
However, for each case, the injection location has been annotated on the images. Because it is on the image we can transform that into CCF space. Most of the analysis you see on the website is with respect to this annotated injection location. Take a look at this SDK document to see how to access the injection site location.