Hello everyone!!
It's been a while, but I thought I would give you an update. I read every one of your post, and I deeply appreciate all your input, concerns, and advices. I took all of them very seriously, and I brought the LVN/RN concern to this lady.
When I completed the hiring paperwork, I wrote RN in the title. Later, I called human resources and asked if they were ok with the title and they said ok. In my first week, the MD's and administration held a meeting to which I was invited. The lady presented me as the new RN who was gonna be running a special project and would also train to be an urgent care nurse.
So far so good. I see that this place might be taking advantage of my need for a job, so they will pay me a lot less than I should get paid, but as long as I get the experience as an RN, the money is not important for now. I can walk to my work, I am getting my hands-on experience, and I still make good money (much better than as a receptionist..

This lady is a good person, which doesn't mean she's not a business woman. When she said she could get me in a NICU setting she meant that she's friends with a GYN MD who works at the hospital I eventually want to work at. She said she could put in a good word for me. Granted, it won't guarantee me a job in NICU, but how can she put in a Good word for me, if she doesn't know me as a nurse? I trust her and so does every one at the clinic.
Again, I thank all of you who took the time to read my post and give me advice. If it weren't for you, I would have risked a lot.