Please allow saving bibliography database per document, not per computer

It’s really annoying having all citations for all documents mixed up in one database.
And if you edit your doc on another computer you lose all your references!

Why is such a terrible system the default

To answer your question … you can save your citations in the document.
Poorly named and poorly documented.
Insert > Table of Contents and Index > Bibliographic Entry > From Document Content > New
This will open a dialog called Define Bibliographic Entry.
Each time you do this it will build the database in the document.
Find previous entries with the Short Name drop-down.

This isn’t a suitable topic for Ask site, where users help each other to solve problems. Your proposal is better placed on Bugzilla as an enhancement request.