Fair enough. I didn't know there is a difference/never noticed it. Its not quite clear to me what the difference is: i.e. How is lifting something here on earth different to the attraction between the moon+earth. I know the calculation is the same...

Wiki has a different take on it too + inconsistent use within each article.

Actually the equations are quite different.

The force due to gravity varies from object to object. The larger the object, the larger the force due to a larger acceleration towards the centre of that object. On earth the gravitational acceleration is ~9.8 ms^-2. On Jupiter that acceleration would be much larger due to it's size. Newtons 2nd law approximates this quite well (F = ma).

Gravitation on the other hand is based on Newtons law of universal gravitation, which states that all objects attract each other with a force of gravitational attraction. The equation for this takes into account the mass of both objects, as well as the distance separating their centres. If I recall correctly I think it's something along the lines of

F = (G*m1*m2)/d^2 where G is the constant of proportionality.