What happens when you check/uncheck the Scene World option, is that blender toggles between the lighting you have set up yourself, and a pre-built (and hidden) HDRI that provides lighting and reflections (as seen by default in the Shading Tab)
So, to get the same look for your render , what you are looking for is the effect of HDRI lighting. Go to the world section of your shading tab and add and Environment Texture (not an Image Texture). You can use any HDRI you want, but the one that is being used in your example is in your blender folder at
....2.92/datafiles/studiolights/world and the image is
forest.exr (there are others there too). You can also find many free ones online - one of the most popular places is HDRI Haven (https://hdrihaven.com/hdris/).
If you want the lighting and reflection influence but don't want the image as your "background", you can mix the Environment Texture with another Background (I used the default) using the Is Camera Ray output of a light Path node as a Mix Factor (as shown below):