Well, libraries are collections of APIs and sdks are usually collections of libraries. So they’re unfortunately kind of interchangeable when discussing them. But I agree with you the correct thing would be to say they’re using Nintendo’s proprietary libraries.
I have a feeling you don’t quite understand what Docker is doing for you and how it works. I suggest looking for an intro to Docker and understand the basics around Docker volumes and networking in docker before trying to orchestrate a complex set of software in Docker.
Don’t give up! I was you about 6 years ago. I’m on my 3rd server setup now, and I’ve gone from where you are now, to being able to script my setup using Ansible and having those scripts versioned in Git, so I never have to worry about remembering how it’s all glued together.