blender 2.81 I press shift+a to add image - background, and then go to render image, and the background image does not show up in the render result.
There's a great addon exactly for this.
Go to Preferences > Addons and search for Images as Planes addon.
Check it to enable it.
Then you can easily add a new image that will appear in your render.
Add > Image > Images as Planes
The way you described is for reference images only - thy're not supposed to be visible in a render.