Is testing a service a functional test or an integration test

We have a .Net component that provides functionality. We have a Restful Web API service that the world will use to call that functionality. We wrote tests that use OWIN to call into our Web API controllers.

In the past, I had always called these "integration tests", because the service is a separate component. However, another developer, who I respect, told me that this is not an integration test, but is instead a "functional test".

I looked at the Defintion of Functional Testing and the definition of Definition of Integration Test and neither one was a clear winner to me for what a test of a RESTful service test should be called.

Is it integration or functional? Are there any authoritative sources that can be used to definitively answer this question (because I don't want my question to be closed for "soliciting debate")?