I would consider myself a runner. I've considered myself a runner since I regularly started putting on my
running shoes and going for a run. Whether it be treadmill or outside, I started by running a mile and a little bit and taking frequent breaks (I could run for about 5 minutes without stopping and weighed 150lbs then). Now I'm training for a half marathon and hey, guess what, still consider myself a runner.
And like so many people have pointed out, everybody starts somewhere. No one was born able to run 6 miles at a time. You work at, just like anything else. You start where you start.
I think it's awesome you're that committed to running and bet you'll be running a full 30 minutes before you know it. Anyone who says you aren't a runner because you don't experience it a certain way is full of crap. Someone once said to me that real runners don't run with music because they don't need it. I say, I can't run without my music and I'm still a runner.
But I do encourage you to try running outside when you're comfortable doing so. It's less physically demanding on the treadmill (I find,), but being outside is a completely different experience. You might end up really enjoying it

.