Defining a month as the time that it takes for the Earth to move 30° in its orbit is likely not practical since it would require frequent changes to keep the months accurate. The dates of apsides, equinoxes and solstices are something better to relocate to an almanac. The same for the number of days of net heating which I have referred to as "summer". It's probably best not to overcomplicate the calendar. One can just as easily draw attention to the perihelion by making it an "Earth day" holiday.
So, what to do about the months? The 7/12ths trick might be useful. Simplifying the day-of-year function would make it easier to compute the number of days between two dates such as the time between one equinox and the next.
When making changes that affect a lot of people it's probably best to adopt a minimalist approach.
The Equinoxes divide the year into two parts. When the Sun is above the Equator there is relatively more warming during this "half" of the year than when it is below it. There doesn't seem to be any collective terms for Spring/Summer and Fall/Winter. But the Earth's heat balance is positive for the former since the temperature rises and negative for the later since it decreases as a study of the Earth's insolation and climate models show. What is often overlooked or downplayed is that there are indeed long term changes over time.