Here is one way to think about it, based on counting days outside Schengen. Some of the corner cases seem to feel more natural when counting this way than when counting days in Schengen.
Every time you spend a day (midnight to midnight) entirely outside the Schengen area, you earn a point.
You also earn a point if you spend a day in a Schengen country that you have a residence permit or valid long-stay (type D) visa from. This also covers days where you enter or exit such a country, as long as you don't pass through other Schengen countries.
Each point expires exactly 179 days after you earn that point. In other words, a point lasts for 180 days including the day you earn it.
When you have 90 or more unexpired points, you can make use of the "short visit" rules.
Since you earn either one point a day, or none, and points last 180 days, you will never have more than 180 points.
What is being in short-visit status good for?
The short-visit rules let you enter and remain in the Schengen area if
You're a citizen of a visa-free country (and traveling with your passport from that country), or
You have a valid short-stay (type C) visa (subject to its limitation of total duration and number of entries) from any Schengen country, or
You have a valid residence permit or type D visa, and are traveling (say, for business or tourism) in one or more Schengen countries other than the one that issued it.
Getting to 90 points
If you haven't been to the Schengen area in the last 90 days, it's clear that you have 90 fresh points. Everything that happened before those 90 days is forever irrelevant -- at least as far as the 90/180 day rule is concerned.
Conversely, if you stay in the Schengen area for 90 days straight and leave on the last possible day, then the next day you only have 89 points left. Since you have left the area, you will now start earning new points, but your old points will expire as fast as you can earn new ones, so it will actually be 90 days until you earn a point that doesn't just replace an expiring one. Once that happens, the previous paragraph applies.
If you have come and left in the Schengen area multiple times, then you need to be more careful counting your points. This can be complex, so this is where the various automatic calculators are handy. The one break you get is that you only ever need to look 180 days back. Everything that happened more than 180 days ago is always irrelevant, as all points you earned back then will have expired by now.
There's no need to (and no point in) attempting to keep track of "when a new 180-day period begins", counting back to the first day you set foot in the Schengen area years ago. (For several months in the early 2010s the law was that you needed to do that, after the European Court of Justice ruled that this was how the old, ambiguous, text in the Borders Code had to be interpreted. But this was quickly followed by a change of the regulation to make the current system unambiguous).
Alternatives to short visits
There are also a some ways to enter and be legally present in the Schengen area without being on a "short visit". These are the ways you can get in legally if you don't (yet) have 90 points:
If you have a residence permit or type D visa from the Schengen country you're in.
You will earn a point a day, as described under (2) above.
If you are in transit on your way to and from the Schengen country you have a residence permit or type D visa from.
You will not earn points on transit days, but your point score will not matter since transit is not a short visit anyway.
If the country you're in happens to accept your presence based on old bilateral treaties outside the Schengen framework. This is available only for certain citizenships, only in some of the Schengen countries. In some countries, having applied for a residence permit will put you in a similar situation until you get a decision.
You will not earn points, so if you want to use this to break to 90-day barrier, you need to put this part of your visit last.
If you enjoy freedom of movement because you're an EU/EEA citizen or a qualifying family member of a citizen.
In this case the 90/180 rule doesn't apply to you, so you don't need to count points. Presumably if you lose freedom-of-movement rights, you can then act as if you had been earning points all the time -- based on a common-sense extrapolation from the rules -- but it doesn't say so in so many words.