Please. It's not a bubble, nor a religion. REST offers some significant objective advantages over SOAP in most use cases. We wouldn't have the proliferation of APIs that we do today if everyone had to use SOAP.

  Instead of the REST finger-wagging, can I have a short
  overview of what functions your API offers? Or maybe an
  example of a request call and its response?

Uh, the page you linked to is an overview of what resources are available, and their content. The first three paragraphs are simply boilerplate to let users know that this is actually a REST API, not XML-RPC, and attempting to head off questions about the example URLs not working.

  So here I am, a developer trying to deploy my WAR file
  on SteamCannon and that’s the API document I find.

Oddly, I don't see anything in that API description about uploading files -- it's all about managing virtual hosts and host disk images and so on.

I assume an "instance" is an actual running OS -- you could scp your files to the instance's public address.

And, with this article, the Bubble Bubble continues to expand...