In the source code of the Rest Services module, I found a comment that gave a solution to the problem. This comment in the method RestConsumer::buildMultiPartEntity says:
// .. or one of its children could be a filedocument. This way multiple
// file parts, or specifically named file parts can be send
Instead of making the main request entity a subclass of System.Image, I had to create a second entity that is a subclass of System.Image and creating an association between my main request entity and the second entity. The Rest Services module then takes the name of that association as the parameter/field name for the file binary data in the request body.
Using hookbin.com, I inspected the POST request generated by the REST Services module and found that the parameter name (i.e. form field name) carrying the image data in the request body is the file name (e.g. "20160617_143958.jpg"), not "Content" as I originally suspected.
So the question is: how can I change the parameter name (form field name) in the request body to be "file" (as expected by the API) instead of the file name?