Is it correct to assume that the relationship between an objects mass and size, is what decides if it’s a black hole or not?
Yes. A black hole is in some ways a very simple object. It has mass, charge, angular momentum. 

These three values determine the size of the event horizon. If the Earth's mass were compressed to less than the size of a cherry it would be inside its event horizon and so would be a black hole.

The gravitational field of the Earth out at the moon would be (almost) unchanged. The moon would orbit an Earth-mass black hole in the same way that it orbits the Earth now.  The small differences are due to irregularities in the shape of the Earth, which wouldn't exist if the Earth were collapsed.

Above the Earth's surface the curvature of space would be (almost) the same if the Earth were to be collapsed. If a rocket were to use its thrusters to hold itself stationary 6400km above the black hole, the inhabitants would experience 1g of gravity, just as we do standing on the surface, 6400km from the centre of the Earth.

If you go far enough below the Earth's surface, gravity weakens, because there is mass above you.  If you collapsed the Earth, then gravity (and curvature of spacetime) would increase all the way to the black hole.