The quick answer is if you keep working on it, then yes some day you will be what You consider a "runner"!  Are you following any kind of training program?  I'm no expert but in my experience the only way to build up endurance is by gradually increasing your running interval(s) and decreasing your walking interval(s).  That, and it does not happen overnight.  I've been running since December, not as many days per week as I should, but finally this past weekend I was able to run my entire Sunday Runday route of 3.22 miles.  I had been stuck at 2 miles for a while but never gave up!  If I can do it so can you, keep trying!

If you think you are a runner, then you are a runner!  Mental attitude is half of what being a runner is about.  I have plenty of friends who I consider runners and who consider themselves runners that run/walk their races.  You shouldn't worry about disappointing anyone, running is something to do for yourself.  Be thankful for the distance you are able to run and keep working on it and you'll find your endurance steadily improving.  Make sure you're setting reasonable goals for yourself to increase the distance you run and you'll be running the entire race soon enough.  If you aren't keeping a running log it might help so that you can see your progress.  You might be running more than you are giving yourself credit for.  Good luck and keep running!

Wait a minute, who are you going to "disappoint"?  Aren't you doing this for yourself?  Maybe the first thing you need is an attitude adjustment.  You aren't doing this to prove anything to anyone else.  You are doing this for YOUR benefit.  If your progress is slow, then at least it is still progress and you are doing better now than you were when you started.

Will you ever be a runner?  Well if you run, then you are a runner by definition ("one who runs").  Will you ever run a marathon in 3 hours?  Maybe, maybe not.  But if your goal is to move forward - literally - a mile at a time then you are achieving that goal.  If you stick with it and push yourself to improve, you will gradually cover those distances faster and easier.  And you will do it all for YOU and no one else!

Oh yes you it will just take time and training.  My daughter, husband and I started running 8 months ago with the inauguration Marathon and Half Marathon in Morgan Hill in November 2010. We felt in love with running right away. As soon as we crossed the finished line we were ready for the next. We have ran 3 1/2 marathons and 2 10ks. On our first half I thought Iwas going to die at the end but my goal was for us to run all 26.2 miles one day and with lots of training we are on track to do it in October 2011 (again in Morgan Hill).  I have read a few books about training to run marathons and I believe that all the information has really helped me with my training. One of the books that i recommend would be The Marathon Method by Tom Holland and like someone else said use the Jeff Galloway method. for every mile you run walk a minute. it takes time,  training and faith but if you really want to complete a 1/2 or even full marathon you will. positive thinking will also help you.  Good Luck and for what I read on your question you are already a runner if you ask me.

I have been running for about a year after power lifting hardcore for a year and  it takes time. I was where you are at for the longest time and then one week it just clicked and everything improved. I found that when you feel like you cant go any further push yourself another 50 feet or another 30 seconds on a treadmill and then repeat, slow down to a crawl if you have to. There is no shame in walking but a whole lot in flat out giving up on yourself.