Actually that is what you asked (although I can see not exactly what you meant)
Q: "On the 96boards product page, there is something identified as release “15.06”. I take that to mean June 2015. Is this the same build as it ships with?"
A:No it is not the same build
If the git tag is in the install instructions (LA.BR.1.2.4-00310-8×16 is a GIT tag) it must have been used for one of the releases.
[jramirez@calypso linux.git (LA.BR.1.2.4_rb1.2)]$ git show LA.BR.1.2.4-00310-8x16.0
Tagger: Linux Build Service Account <lnxbuild@localhost>
Date: Tue May 19 05:55:34 2015 -0700
Merge: bf19b76 88b27d9
Author: Linux Build Service Account <lnxbuild@localhost>
Date: Fri May 15 07:33:38 2015 -0700
Merge "arch:arm: msm: removed not needed device trees" into LA.BR.1.2.4_rb1.2
Please point me to the instructions you are following and maybe reword your question so I can help.
I am really not sure what you are asking: If you know how to update everything it means you already have access to all the necessary repositories and therefore it is a matter of identifying the commits.
Are you perhaps trying to learn how are the releases generated?