The Satipatthana and Mahasatipatthana Suttas strongly informed Mahasi Sayadaw's noting style of insight meditation, which is very specific about how to practice in accordance with the instructions in the sutta both on the cushion and off. An excerpt from the "Basic Practice" section of Practical Insight Meditation:
When you look at the tap or water-pot on arriving at the place where you are to take a drink, be sure to make a mental note, looking, seeing.
When you stop walking, stopping.
When you stretch out the hand, stretching.
When you touch the cup, touching.
When you take the cup, taking.
When dipping the cup into the water, dipping.
When bringing the cup to the lips, bringing.
When the cup touches the lips, touching.
When you swallow, swallowing.
etc.
From how I understand this tradition's teachings, the way to correctly practice according to the Satipatthana Sutta is not to specifically try to practice what's in one section or the other, but to continuously mentally note everything that occurs at one of the six sense doors, until you're able to see not just the thing that happened at the moment it was sensed but also when it arose and when it perished, while always returning to the rising and falling of the breath at the abdomen (or in and out of the breath and the nostrils, if you prefer) as an object when there's nothing else going on to note.
That's in contrast to your questions, which ask about practicing specific parts or sections at a time — even if you try to only note the breath, thoughts will arise, sounds will arise, etc., so this method recommends noting them all and then returning to the breath, rather than trying to exclude them.
