I'm posting a summary answer just to clear the question, so it has an "answer."
First, I think my original question was off target to the final answer. I wanted to notify a "parent" buffer when a child buffer that it spawned was killed, so the parent could refresh the buffer contents when it regained focus after the child (temp) buffer was killed. So my focus was on finding out how the parent buffer could figure out that it just became active.
There was not a easy way to find that out, or I suppose someone would have posted an answer.
So what really worked was having the child buffer call a defun (a hook function) that just updated the parent's contents. At first I used the global buffer-kill-hook, but that led to problems.
First, my hook function had to protect against every other killed buffer running the guts of my hook function. So I modified my hook function to look at the filename regexp of the killed buffer, so I could spot the filenames that were normally child buffers.
That was ok so long as only the parent edited those files by spawning them into a child buffer. But when I edited those special filenames through dired and killed them, my hook function tried to update a parent buffer that wasn't there.
When Stefan pointed out the "local" argument to add-hook, I used that feature to have the parent install a buffer-local hook on the child buffer, so that my hook function could only be run by a child who was spawned by the parent and who had the local buffer-kill-hook installed by the parent.
Now I can edit the special files through dired without bad effects, since the buffers don't have a local buffer-kill-hook installed by a parent, and because I'm not using the global buffer-kill-hook.
So everything is working fine now. Thanks to everyone for helping.