While I don't disagree with the general sentiment on this topic (other than
Jim Green's pontifical declaration of what IS and ISN'T) let me throw out
the following:
Weight = mg if taken as an 'operational' definition will then have 'g'
to be the instantaneous free-fall acceleration (in a vacuum) of an object
of mass m due to the influence of a much larger mass (such as a planet) at
a particular point in space relative to the large mass. To be even more
precise: Weight is the term used to describe the gravitational force
between a very large mass (such as a planet) and other much smaller masses
(usually at or near the surface of the large mass), that would cause a
free-fall acceleration of the small mass of magnitude 'g' towards the
center of mass of the large object, in the absence of all other forces.
Quite a mouthful, but the point being that if viewed from an operational
framework, then the 'g' in the definition of weight IS still an
acceleration even though no ACTUAL acceleration need be occurring. The
field explanation of 'g' is a different, more compact, and more
mathematically satisfying way to look at this, but the operational
definition approach seem to me to be just as valid.