Have you ever run a marathon running race? After a gap of over 30 years and at the age of 56 years, I ran my first five kilometers run in Athens, Georgia, USA along with my daughter.
Is relaxed easy running better or fast sprint running better for the running event? This was my question at that time. It was my first run and eventually I could complete it only with a mix of easy running and walking and it took me almost 45 minutes to complete the five kilometers. The first prize winner of that event did the entire run on a fast sprint and finished it in under 15 minutes. I went on to run several five and ten kilometer runs and even won the second price in a ten kilometer race in my age group category in Hyderabad. I learnt that we need to begin the race with a very slow easy run and then run almost the entire distance using a rhythmic body movement and with eyes fixed on a spot a few meters ahead on the road and doing conscious breathing or active relaxation breathing as you call it. The last one kilometer or so can be covered with a fast sprint. As one becomes a pro at running the sprint portion can be progressively increased.
With meditation it is the same. Always begin the meditation with eyes open fixed gazing and conscious breathing. I call this FGCB and it is explained in the linked article below and it is probably same as your active relaxation breathing. This will gradually balance the energies and settle down the restless thinking process of the mind. Breath watching is possible only after a shift occurs in the state of the mind. I call this shift as grounding. The vibrations of the electromagnetic field around our body, our biofield energy, connects to that of the earth and starts vibrating at that frequency and the mind becomes quiet. I actually recommend doing at least 3 minutes of energy movements before sitting to meditate. If you are interested to learn about this please read a few of my other posts and answers.
Breath watching is also just a stage and eventually deep meditation is simply being. The thinking-mind is quiet and the aware mind is alert, awake and aware. You are simply being. Being in a state of non-doing being. Watching one specific occurrence is also a subtle form of doing. Complete non-doing being is a witness state of acceptance and allowing. The aware mind is simultaneously aware of everything that is arising in the here and now, within the body and mind as well as in the surroundings but continues to be a disengaged witness.
To answer your question, active relaxation breathing is better at making the restless mind settle down and be quiet, breath watching is better for sitting and practising with, until it too is no longer needed for sitting and simply being.
FGCB is explained here:
What are some breathing exercises for beginners who struggle with focusing on their breath while meditating?