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Sorry I don't have much time today to get into it. Seems to me you can't solve for case 2 here since in this case m2 and m1 are switched. But it does not matter, I am not trying to solve for speed before and after.
The force of the impact does not depend on the mass, I agree, but the energy to dissipate (in the cyclist body) is much higher. I'm just saying that inertia plays a role as it contribute to the energy necessary to stop either vehicule. I am happy to be proven wrong, I just don't think this is the right equation to do so.
No, for both cases, body "1" is the biker and body "2" is the car.
This is just for the sake if simplicity. The force does in general depend on both masses, not just the mass of the car. Yet, the biker has only ~ 5 - 10 % of the mass of the car and thus, their mass can be neglected and the simplfied solution (m_1 / m_2 -> 0) doesn't include masses anymore
Exactly. This part is included in the coefficient k. Yet, for the simplified solution, the biker doesn't stop the car in any form.
Suppose a completely plastic impact, k=0: The biker would be stopped to zero velocity in case 1, and in case 2 they would be accelerated to the velocity of the car. Here the magnitude of the force/acceleration doesn't depend on whether the bike did move or the car did move.
For the elastic case, k=1, car and bike are treated as billard balls: For case 1, the biker moves with the same velocity as before, but in opposite direction. For the other case, the biker would move in opposite direction, but with the double velocity as in case 1. Thus, here, the force causing the acceleration must also be twice.
So as long as the impact is not purely plastic, it does matter whether the biker hits the car (case 1) or the car hits the biker (case 2).