The "Save Project" & "Save Project as" options will not export a part. You can use these to save your work in the editor so that you can continue editing the part later. These buttons will create .8proj files, which can be opened in the Editor to pick up where you left off.
To export a part, click "Generate Part" from the file menu. Name the part if you have not already, and then select a directory to save the newly generated part.
After a part is generated, it should automatically appear in the parts list of the User Library. If your new part is not appearing in Soundpaint, you can click “Manage Library & Part Directories” from the menu in the top left corner, double check that the folder where you saved the part is on the list of directories, and click “Full Rescan.”