Your English is little hard for me to understand, but let me still try to answer.
According to what I think, gravity is accelerating myself no matter what, and what it's actually preventing me from going down it's not the normal force, but the friction my body experiences against the surface which it's total, this is the reason then why I'm at the bottom of the atmosphere.
Normal force is force produced by the surface on the object in perpendicular direction. Friction is force produced by the surface in tangential direction.
Well as think of it, friction is the resistance an object experiences by another object when put together. Normal force it's the contrary force an object experiences
You are basically arguing about linguistic, not physics.
First of all I do not see that much of a difference between your definition of friction and your definition of normal force. One is contrary force an object experiences, another is also contrary force an object experiences with few more addendum. So from your definition every friction is also a normal force.
Second and more importantly - as I said, normal force and friction are both forces the surface exerts on the object. The difference is in the direction. As the force which is countering gravitational force is in perpendicular direction to the surface, this force is normal force and not force of friction. If the object would be on a slope, then both normal and frictional force together would counter gravitational force.
The main point of this division is due to the different properties of these components. Normal force is there to just cancel the perpendicular motion. It is force of constraint.
The static frictions is there to cancel tangential force acting on object at rest, if it is up to the task. That is, once the tangential force is big enough, it overcomes static friction and object gets into a motion.
Now, the dynamic friction acts and it is simply a force which resists the motion of the object and tries to bring it into a rest w.r.t the surface.
All these three forces have different formulas and must be tackled differently during calculations and therefore we distinguish them. Their true nature is however quite similar and one could argue that from philosophical point of view, there is just one force which surface exerts on the object.