After doing more digging, it seems like Sitefinity generates the ID of the search box on the fly. So in my example:
the "Search" part of the string is the ID for my placeholder
the "TCE5DF3E6002" is the ID of the control generated by Sitefinity
and the "searchTextBox" is the ID of the textbox control the user is actually inputting their search term into.
So, the control ID "TCE5DF3E6002" seems to be what I need to be able to programmatically retrieve. The question is, how?