Database system vs database management system
Most are saying they are the same.


They are the same, pretty much. Most people use the two interchangeably.

You could try make a case for "database system" to cover simpler arrangements where there is just a database and not management tools (for instance in memory database libraries and/or like sqlite is often used), but there are so many grey areas here defining the difference is going to be difficult to do in a consistent manner.