I pointed out you likely need to look to du Châtelet to discover the derivation of what Coriolis refers to as the "old usage".

As to the "1/2 is a mistake", one question in physics is, is gravitational mass exactly equivalent to inertial mass?

You can omit the 1/2 and continue with the old definition. If you do so, then your system of physics will have an inertial mass which is half the gravitational mass.

As such, it's a difference only of philosophy. The main advantage to the 1/2 is the math is cleaner if we assume inertial mass = gravitational mass.