Gravity: Newton vs. Einstein. Do they both define the same phenomenon?

Gravity...What is it? That's the million dollar question isn't it! As a kid in school you were most likely told that gravity is an attractive force between bodies with mass, and as far as I know that answer still holds true. So why do we also hear that Einstein overturned Newtons theories on Gravity? This is a question that I have asked myself for some time now. In doing so I have come to a few possible conclusions. Einstein's theory of General Relativity states that bodies with mass warp the fabric of space. It is almost as if a large body 'displaces' space as it would with water. This is an interesting thought. For instance, it implies that the Earth doesn't pull us towards itself. According to Einsteins theory, it is space itself that pushes us towards the Earths surface. In a sense, Einstein said that gravity isn't the result of an attractive force between bodies with mass, but that these bodies warped the fabric of space and it was this warped space that has an effect on the other surrounding/passing bodies. This is all fine and great, but really, how does Einsteins version/concept of gravity explain the effect that our moon has on our tides? It seems clear that the moon pulls on our tides, i.e. our moon has a Newtonian attractive force on our tides. So what does this mean? In a nutshell, two separate theories exist. Both theories make verifiable predictions. Both theories are sound but describe two separate phenomenon. Can they both be called gravity? It seems apparent that bodies with mass do in fact warp the fabric of space, and at the same time attract other bodies due to some invisible, yet calculable force. We just lump it all together and call it all gravity. I believe that gravity needs to be seriously redefined...That is if we can ever figure out what it is! If you have any graduate level astronomy or physics knowledge I would love to hear your replies on this.

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