I'd say that despite the non-standard terminology, that first extract is clearly talking about 'reaction' forces. These are 'real' forces due to the interaction of two different bodies.
That second extract, despite its failure to make explicit the fact that it is using a non-inertial frame of reference, seems clearly to be talking about inertial forces. Note that the applied force and the inertial force both act on the mass in question. (Not so in the first extract.)
Apples and oranges, I'm afraid.