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A container is the general concept of creating a sandboxed run environment and can be used as a general term to refer to either Docker or Singularity images. However it is sometimes used to also refer to the specific files being generated. This is probably not ideal, as it can clearly cause confusion to new users.

image is generally used to to refer to the actual files created by singularity build ...

instance refers to a specific way of running singularity images. Normally, if you singularity run some_image.sif or singularity some_image.sif some_command you can't easily access its environment while it's running. However, if you instead run singularity instance start some_image.sif some_instance1 it creates a persistent service that you can access like a docker container. The singularity service/instance documentation has some good examples of how instances are used differently than the basic exec and run commands.