Library Colour Swatches not Retain Names when Added to Swatches

Yes, swatch libraries are local, saved on your computer and loaded using the Swatches panel.

Colors added to a CC library are stored in the Cloud and synced when opened in Illustrator.

But Spot colors, Global colors, Pattern swatches, Gradients and Brushes are not very well supported in CC libraries.

If you first save your gradient to your swatches panel, it will be automatically added there in new documents if you follow these steps.

But as a workaround you can drag an object using a brush/pattern/gradient/global color to the CC Library

You can also drag multiple items using different brushes/patterns/gradients/global/spot colors together into a CC library to simulate a library. Text objects must be grouped before adding them as graphics.

The easiest way to add them to a new document is to hold down the Alt key when you drag them from the library and place them in a new document. The patterns/gradients/global colors automatically get added to your swatches panel.

You can hit escape before you click to place the object(s) or delete the dragged library items because the swatches and brushes are now part of the document.