Isn't it equally plausible to say that the earth has accelerated from the point of view of the reference frame of the car?
It is! Just as the motion of the planets has once been described by the epicycle theory (which was very complicated compared to heliocentric theory), you can consider the earth accelerating towards the car. Choose whatever reference frame is suitable for you, and be prepared that other people have other, more suitable descriptions, which allow them to make predictions in less time with less words than you.
Actually, from an economic point of view, neither the car accelerates towards the resting earth, nor does the earth accelerate towards the resting car. It is that car and earth are accelerating relative to one another, inversely proportional to their mass. This is because that description allows for the usual angular momentum and linear momentum of the isolated system "earth+car" to be conserved, which answers
Are the forces being applied to move the car have a special significance in deciding which frame is inertial and which one is not?
The forces that act between earth and car are only significant with respect to inertia, in that the sum of all forces acting on the system "earth+car" are zero (implying conservation of linear momentum) and the sum of all torques are also zero (implying conservation of angular momentum). This defines the inertial system in classical mechanics.
In general relativity (gravitational field theory), things are getting more complicated, but you wanted a simple explanation.