You should do some more study on database technology.
A database system acts as a black box, a mediator between the user or calling programmer and the data being managed by the DBMS. The actual data persisted to storage is never exposed to users.
So there is no need for shared file storage such as NAS. The DBMS should have its own private directly-attached storage for safety, performance, security, and simplicity/reliability.
The DBMS handles concurrency, juggling simultaneous access by multiple users. How they do that juggling varies, such as optimistic locking in a scheme like MVCC or simple pessimistic locking. You must learn about the various approaches to learn about what works for your needs.
SQLite is meant to be an alternative to storing data in files, not intended to be a serious database server.
As for MySQL and its forks… I suggest you instead consider Postgres, the world’s most advanced open-source database system. But actually, for only 10 users and only 10,000 records, most any database server will do. The H2 Database is another possibility.
A database server only stores your data. To access the data you either need an admin tool or you build a custom app.
There are many many admin apps out there. But doing data entry through them is a bit awkward and clumsy, similar to using a spreadsheet. Could work for very simple needs.
To build a custom app, you have a whole lot of learning ahead. Personally, I use Java with the Vaadin Framework to build the user-interface as a web app. But there are many other routes.
As a beginner, I suggest you consider FileMaker and 4D. Both have a simple data-entry and reporting facility in their own client app. And both offer an app development environment integrated with a proprietary database server. Both are relatively easy to get started with for a beginner but are also powerful enough for use by professional software developers.