Shadow Catcher is a generic art term meaning create something to catch a shadow. They have been used in animation from the very first 2D animations. Techniques and methods vary and get even more complex with variety when trying to do them in 3D. You can use Blender Cycles and create one using Blender Shadow Caster and export the shadow as an image and texture it to a plane and use it. You can use the EEVEE and do the same thing. You can also try to create a material…lots of options but the general term is Shadow Catcher.
The effect you present in your image a2. Could be done with a simple light and plane as a background and you would bypass the entire need for a Shadow Catcher. Lighting may be something you don’t want to use due to performance issues but if your application fits…that might be the better option based on the graphic you provided.
The sample image you also provided at the start with the robot can easily be created with a plane and an alpha texture under the object. You will notice there are no shadows for the robot arm…just a generic circle under it. You could do the same thing for your object as well.
Render out your object with a transparent background. Bring it into your photo editor and desaturate to make the object black. Gaussian blur it to create a shadow. Save your image as a PNG or other alpha-supported format and apply to a plane.
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