Mass and acceleration

I'm confused about how the two phrases; 'Acceleration is independent of mass' and 'Acceleration is inversely proportional to mass' can coexist :/

Thanks in advance!

In the equation F=ma then a=F/m which means that for a constant force, the acceleration is inversely proportional to the mass it's applied to.
So far so good.

In the case of the acceleration due to gravity, however, all masses accelerate with the same amount of acceleration. (This was Galileo's famous experiment)
So we say that g is independent of mass.
The reason for this is that for a larger mass, there is also a larger force. A mass that is twice as large as another, also has a force of gravity (its weight) which is twice as large as the other. So it finishes up with the same acceleration.
g=F/m for the 1st mass and g =2F/2m for the one with double the mass (and double the weight).