Well, its semantics right? Gravity causes acceleration, so its a force? It can be treated mathematically as a curvature. It behaves differently in different frames, but so do electromagnetic forces.
If someone applies a force to you, you can feel it. If you're in free-fall, you can't feel anything. According to general relativity, there's no way to tell if you're stopped, moving in a straight line with constant velocity, or in free-fall in a gravitational field. They're all the same.
Actually, you only feel it if the force is applied differentially (which most forces we experience are). If your body had a volumetrically uniform electrostatic charge, which was not high enough to have charge screening, and you were acted upon by an electrostatic force, you wouldn't feel anything either. It is just that 'mass charge' is always distributed evenly along with mass, while with electrostatic forces this is a special case.
Forces can indeed be modeled as a curvature in space-time. However, there is a bit more to GR than just that, which is why it took Einstein years to go from SR to GR.
There is a distinction. A gravitational field has a center, so if you are in free-falling towards it you can determine that you are in it.
Take two objects and place them apart to float stationary( the line they make should be orthogonal to the center of the field ). Even though they started completely stationary, they will slowly start to move towards each other, as an unknown force is acting upon them. In reality they are falling towards a common center.
I guess this doesn't work on dimensionless points, so the your comment is correct as far as mathematics are concerned.