OK, definitions:
"G", the big capital G in the context of aviation is the acceleration in a direction perpendicular to the movement excluding gravity. In other words, it is the force over mass, where the force include the sum of all forces on the plane except gravity and then taking only the component perpendicular to the velocity.
Since drag is by definition along the velocity direction and thrust is usually close to be aligned with the velocity and we exclude gravity, the only thing that is left is the lift which by definition is perpendicular to the velocity. Hence G=L/M, but expressed in units of earth gravitational acceleration, usually denoted with a small g. So if you use SI or CGS units G=L/(M*g)=L/W where "W" is the weight.
Some finer points:
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* Generally, G is NOT the actual acceleration of the plane in any axis or total since G does not include gravity.
* a stone being thrown is at 0G from the moment it leaves the hand till it lands, even while going up - the only forces that act on it are drag, which is aligned with the velocity and hence irrelevant to G and gravity which is explicitly excluded from G. Dignify all of us by not arguing that the stone is not spherical or is rotating.
* The pilot feels G by how hard the seat is pressed against hit butt, not his back. In a 90 degree climb (maintaining the angle), the seat is only pushing against the pilots back and not his tuches. The vertical acceleration can be whatever, but since it is aligned with the velocity it is still G=0.