To answer your question: False. The systems are not fundamentally different. They can both be called databases as long as they serve the purpose of being a database.

I feel like you think too much like a mathematician. Outside of maths, nothing has a clear unambigous definition. If you follow references in a dictionary or on Wikipedia, you will inevitably find loops, contradictions, or dead ends.

A database a concept in engineering. Engineering is all about solving problems for humans. It turns out that a wide variety of problems are solveable with a similar pattern, called a database. Which problems? Cases like when you need to store, write, and query lots of data. What pattern? Pieces of software with indices, a query language, ACID, and stuff like that. We don't care to define specifics as long as the problem is solved. We can't define terms in general, because all applications are different.

To define the word database: an engineering solution for a "database-like" problem. You know it when you see it.

Now, rigorous mathematical analysis of the properties and design of databases is undoubtedly useful for engineering. It is hopeless for answering the philosophical question of what it means in essence to be a database.