You just need to be careful about the distinction between certain individual interactions (forces), and the net force on your body.
Newton's Second Law demands that the net force on your body is your mass times your acceleration. Your acceleration is zero when you're sitting still on Earth because the net force on your body is zero; the gravitational force pulling you downward balances the normal force of the ground pushing you upward. This does not mean that you can't feel the the normal force itself.
You will be able to feel any such contact force, even if the total force on your body is zero.
So here, really, lies my question: Is there even a point to arguing about this?
Perhaps there is a point in discussing this.
In the Newtonian point of view, impulse and change of momentum are different concepts. Why?
Force $F(t)$ is a basic quantity describing instantaneous influence of one body on another, in general having a magnitude and direction, but let's have everything in the same direction here for simplicity. The formula for force has to be inferred from other laws of physics - it can be due to gravity ($mg$), spring ($-kx$), or air resistance $(-cv^2)$ or others or their combination.
With this notion of force, impulse of the force $F$ in the time interval $t_1..t_2$ is defined as
$$
I = \int_{t_1}^{t_2} F(t) dt.
$$
One could calculate impulse without knowing anything about momentum.
Now, based on the 2nd law for body with constant mass (not definition here)
$$
F = m\frac{dv}{dt},
$$
we can derive that
$$
I = m \Delta v
$$
and since $m$ is constant, also
$$
I = \Delta (mv).
$$
which means that impulse equals change in momentum. This is the historic and common point of view, I believe.
Alternatively, if you take the point of view where "force" is defined as $ma$, then impulse and change of momentum of the body have the same values as a consequence of definitions only. But I wouldn't say this means that impulse and change of momentum are the same concepts, because they are introduced in a different way with different name and symbol. So either way, I think it is safe to say that both are different concepts, while having the same value, either approximately (if 2nd law is taken as approximate law of physics) or exactly (if it is taken as a definition of force).
Best Answer
The position you are taking seems to depend on hindsight. Put yourself in the position of Newton being the first person to state these laws.
The first law was a flat-out statement that Aristotle was wrong when he stated that "nothing moves at all, unless a force which causes it to move is acting on it." Of course everybody now "knows" that Aristotle was wrong about that, so the "shock and awe factor" of Newton building his entire argument from that starting point no longer exists.
The second law then gives a definition of how to numerically measure the notion called "force." Of course it is consistent with the first law, since common sense would say that "no force" must have the measured value of $0$.
In modern terminology, the third law is a statement of the principle of conservation of momentum. It is independent of the first two laws - and apparently, the many crackpots who are still trying to invent perpetual motion machines and "free energy" devices still don't believe it is true, despite the empirical evidence (not to mention Noether's theorem).