During meditation i notice that the mind tends to alternate between one-pointedness (breath touching the tip of the nose while breaking off any upcoming thought) and classifying upcoming thoughts into thoughts about the past, future and related to ego, before breaking off the thought, or observing the upcoming and cessation of all sensations in the realm of body and mind while always keeping adhithanna in posture. Is this inconsistent or can a unified and one-pointed mind exist alternately or even simultaneously?
It's either one or the other. One-pointedness means exactly that - the mind is fixed in one place. Anything apart from that single point detracts from concentration. What can happen is that as you develop one-pointedness, your attention can get somewhat "smeared". You'll still have some hold of the single point, but your attention isn't fully fixed on it. In this case, your vitakka or applied thought is [mostly] there, but your vicara (e.g. sustaining thought) is either slipping around or is intermittent.
If you find yourself moving away from single-pointedness on a fixed object, my best advice is to switch to something a little less subtle like the full inhalation and exhalation as it travels through your body. It is much easier to maintain sustained attention on something that is moving. In a sense, the motion keeps your interest more than would a single point. As your concentration deepens, you can begin watching the where the breath makes contact again.
Bonus point - don't watch the breath, wait for the breath. What I mean by that is that your attention should be passive as if you were waiting for a train to arrive, a gaggle of geese to cross the road, or your Hot Pocket to cook in the microwave. Westerners especially can be overly aggressive when they actively try to watch the breath. The pin the poor thing down with their efforts and don't give it any room to...well...breath. As the Hsin Hsin Ming puts it, "when you try to achieve passivity, your very effort fills you with activity". Strong arming your attention like that arouses needless agitation. If you take a more patient approach, your concentration will be correspondingly deeper.