First of all, try to remember that meditation is an extremely personal affair. The best approach for you may not fit exactly into one of the many popular methods.
That being said, I will share my perspective based on my personal experience:
In the beginning of the practice, you will most likely be "forcing" the mind to stay on the breath since you will not yet have learned how to gently direct attention. Over time, this forcing will fade into the background as the mind becomes capable of lightly putting distracting thoughts to the side. Once you have developed a decent amount of insight, you may reach a point when this nudge towards the breath becomes so light that it is not even noticeable. This is because you have weakened the attraction to those thoughts in the first place; remember that until the attraction/obsession with constantly thinking (rather than knowing) is weakened, any attempt to direct the mind will be met with resistance.
So I suppose in my opinion, both methods lead to same results since if one starts with "choiceless awareness" then one is learning to gently step back from the constant internal yammering, and eventually gets to the point where the attention very easily rests on the breath.
With the other approach, directly attempting to redirect attention, you are learning to recognize clearly when the mind has become distracted from one's current "goal" (the air quotes since if one becomes obsessed about this goal, it can lead to serious issues). This eventually leads again to a similar clear awareness of whats going on in one's mind as it is happening leading once again to an eventual weakening of obsessive thought. Finally as before, the mind now has the ability to rest on the breath without force since there is no longer any longing for the thoughts that might have previously forced one to use tension.
Overall just try to be gentle with your mind, while at the same time cultivating a bright awareness.