I moved in a residence on March 30, 2013 orally agreeing with the elderly landlord to pay $600 a month for rent a room. On June 5, I owed her a balance of $100 for May 2013 rent and $600 for rent for June 2013. She gave me a 2 week Notice to Vacate on June 5, 2013 to be out by June 18th (two weeks). On June 8, 2013 I gave her $100 balance for May rent. She said she will not take any more money from me, although her Notice to Vacate stated Failure to pay rent. Can she refuse the money and do I have to move with only a Notice to Vacate?

Once you are late you are in breach of lease (without a written lease it's a month to month lease)... they don't have to take your money at that point... and in fact by taking it would hurt their effort to remove you.

She can even take your rent payment and evict you for being late with it. You seemed to be getting lax about being on time pretty quickly, only paying one month (April) in full and on time? It's to be expected that a landlord would not want to continue.
And she can terminate an oral arrangement the same was as a written one.
If you like the place and think you can be more diligent about 'rent comes first' in life, then try to apologize and promise that you will turn over a new leaf. But it might be too late. You are supposed to be out tomorrow.

I suggest you check with the County. There seems to be a distinction between tenant and boarder in your county. Though I couldn't find the definition of boarder, but a boarder has less rights than a tenant.

Usually a 15 day notice would be insufficient, but it may be legal because you are considered a boarder. No, you don't have to vacate by the deadline, but then the landlord will have to go to court for an eviction order.

You seem to think that chronic late payments are OK. The landlord doesn't have to put up with it. You were late 2 of the first 3 months of being there. I wouldn't put up with it either.

You should just move, since they will be able to evict you, and you have given them ample reason to do so.

Rent is to be paid on time, and your statement only shows they have cause to do so.
I would have evicted you from any of my rentals also.