Differences between Quaternion integration methods

Method 1: As was pointed out the first method uses the angular velocity to calculate a difference-orientation $q_r$, which accumulated as a result of the contant angular velocity. This method is an Euler-approximation of the integral of the angular velocity! $$\Delta \alpha = |\vec{r}| \Delta_t$$ is a first order approximation of this difference angle! In the next step this angle is transformed into the quaternion $q_r$ and then this difference rotaion is applied to the current rotation.

The benefit is a correct integration for constant velocities. The drawback is a computational effor by using trigonometric functions.

Method 2: Here, the calculations are ideal up until the derivative of the quaternion $$\dot{q} = \frac{1}{2} q_r \, q $$ which has no simplifications in it. Now at this (later) point you apply the approximations that 1) the angular velocity is constant and 2) that you can integrate this quaternion derivative by a first order method. You can easily use a higher order method for integrating this last equation, yes! RK4 is a valid choice.

Conclusion: Both methods include a first order integration. However, the first one yields correct rotations for constant velocities and is affecting the norm of the quaternion less than the latter approach. However, using a higher order method I would prefer the second approach, since it is computationally more efficient. This is the main point of method 2! You can implement this on really cheap hardware with high speed even with fixed point arithmetics!