I may be wrong, but I don’t think there is one. The term is generic. There are good and bad definitions out there. This seems very reasonable.
You’d actually need to define what a database is and then look for a distinction. What most people think is a database is probably too specific a definition and applied only to a part of what might be considered a database. The very definition found in Wikipedia is pretty bad because it tends to a very specific form of database.
It’s not really a term, but if you want to name it, call it "program for accessing/manipulating databases". Who knows "program directed to database"? Depending on the context (does not appear to be the one used in the question) it can be "database manager program", or simply "database program" (database software).
I think the name will depend more on what he actually does than because he accesses the database. It is difficult because, in general, access to the database is even a secondary activity, it is a means to reach the end. And what’s important to define the type of program is its purpose.
Although it can be very strange and almost unviable, some programs that are always expected to work with databases can work without them and reach the same end.
I prefer to give a name of ERP, CMS, DNA Analyzer, or something like that, it’s more significant than saying that it accesses database.
The "database manager thing" (database manager) would be a program whose end activity is to manipulate database - possibly to assist other programs - so it is easier to give a name to it. I don’t even know if that’s a good name, but it’s pretty used.
It is common, but not mandatory, that more than one program is responsible for manipulating a database. Of course more complex databases tend to this.
Note the difference between program and solution/complete system.
Well, that seems to me to have little relevance. It may seem precious what I am answering, but just find a term for it only fits to the most academic field even.
Anyway it would be nice if someone knew something more specific, I doubt.
"Database-backed software" and "database-driven software" are ways of saying software supported by database and database-oriented software, respectively.
– Piovezan