Like you (I think) I get stuck in terminology. I am a very literal person. Laurie takes great care to be precise, but in this environment of three or more apps trying to work together, some terms get duplicated.
As I understand it, the "alert database" in this case is the database within JTAlert that keeps track of what you want to be alerted for.
When we "rebuild the alert database" (by the way, terminology for this action was recently changed to better describe it) we are scanning the database of the logging app, to see what we have confirmed or not "as of right now."
This should be done after any new contacts have been confirmed in the log, such as after updating the log from LOTW. Then for example, JTAlert will see that we no longer need a country or prefix or whatever you are looking for, and no longer bother you with alerts for stuff that you have confirmed.
Hope this helps.