I am curious about where you got these "definitions" from? I have never seen either of them described that way. As far as I know people with DID used to be called "multiple" because the name of the disorder back them was multiple personality disorder. But now it is referred to as Dissociative identity disorder instead. My guess is people who were diagnosed before the changeover might refer to themselves as "multiple" but I have never seen that as denoting the person to have multiple parts but be high functioning. Equally I think people who are diagnosed with DID can be either high functioning or low functioning or anywhere in between.

I say I have DID because that is what I am diagnosed with. However I have always held a job and been able to support myself and I have never been in hospital. I consider myself to be high functioning even though I definitely do have a disorder that seriously impacts certain areas of my life.

I'd like to add that I see myself as psychologically injured rather than as having a mental illness. I have a psychological injury due to trauma.