Ok, I see. I paraphrase for your confirmation.
When an object is standing on the Earth, there are two interactions: one attractive (gravitational) and another repulsive (electromagnetic = normal).
Each one has its own pair of 3rd Law forces. The 3rd Law pairs (acting on different objects) do not counteract each other.
Hence it is the mixture between two forces of different kind and opposite sign (which are not 3rd Law pairs and do act on the same object) what causes the equilibrium, isn’t it?
Gravitational Force by the Earth on the object (pushing the object downwards) is cancelled by Normal Force also by the Earth on the object (pushing the object upwards).
Gravitational Force by the object on Earth (pushing the Earth upwards) is cancelled by Normal Force also by the object on the Earth (pushing the Earth downwards).
The fact that the forces that cancel against each other have equal magnitudes is not due to one being a reaction against the other, as noted, but to… the fact that we have assumed that they do cancel against each other. In fact, when a ball bounces off the floor after being dropped from some height, they do not cancel: in that case, normal force prevails at collision time and hence the bouncing.
Some language problems:
Usually the gravitational force is referred to as weight. But in common language one tends to think that weight is that sensation that arises when weight combines with an opposite force that makes it “apparent”. So people think that whenever that sensation is absent, like in free fall, the body is weightless, which is not technically true: it is only “supportless”. Hence people should stop saying that or… physics could change its nomenclature and reserve the word “weight” for “apparent weight”. As the latter is too ambitious at this stage, your suggestion is at least try to avoid the word “weight”, to avoid discussions... Good.
Usually Newton’s third Law is explained saying that for every action, there is a “reaction”: a force by A on B triggers a “reaction force” by B on A of the same nature, same magnitude, same direction and opposite sense. But in common language we use the expression “reaction” also to refer to situations where a force by A on B triggers and opposite force also by A on B. This is our case, where gravitational force on the object exerted by the Earth triggers, when the object touches the ground, a reaction through another force, normal force, exerted also by the Earth itself on the same object. Who should rectify here? To me, physics should not use the word “reaction” in the context of 3rd Law. The two sides of a 3rd Law are really the two sides of the same thing. It is an interaction, a single phenomenon, having effects on both sides… What would you advise?