"Dose absorbed" is a physical measure of ionizing radiation that is directed at something. That's measured in rems or sieverts (or Grays, notice the spelling). "Dose equivalent" is a medical measure of the risk caused by that "Dose absorbed". That's measured in Greys (with an E, not an A). Both those measures combine the count rates for the 4 different types of radiation into 1 amount. "Dose equivalent" goes farther and is meant to calculate medical risk to a living person. Even more confusing there is a Gray (dose absorbed) and a Grey (biological impact of a dose absorbed, or dose equivalent); they are different.
The part about beta and gamma radiation is about establishing a baseline for converting between the two units but should never be used for calculating "dose equivalent" in practice. Its how we determine the value of 1 Grey. It isn't a way to convert from "dose equivalent" to "dose absorbed".
I'm trying to simplify this stuff for an audience without the necessary background information. Doing that requires me cutting out a bunch of details. The NRC text on the other hand is technically precise but also is mentioning a lot of things that are true but confusing or only useful for calibrating instruments. They are also explaining one tiny part of this and you left out the parts where they talk about how to combine the different counts for each type of radiation into one measurement. That's something else I'm also trying to explain at the same time. So I'm covering more ground and trying to do so with simpler terms. That's going to mean you can cherry pick stuff but doing that will to give you the wrong impression.
Natural background radiation varies by location on earth by a factor of 300. That 4 mSv per year is for natural background radiation at the low end of the scale which happens at sea level in places without Uranium or Thorium deposits. However, there are places where people live (and have lived for 1000s of years) where the natural background radiation is 300x that amount or about 1.2Sv/year. There is no observed increase in cancer rates in those locations despite decades of study. I'm also assuming that the "normal person" will take a flight or two and potentially be near other sources of radiation without knowing it (like your smoke detector).
PS The 50 mSv/year number is absurdly low. Its one of the main complaints about how the NRC handles nuclear radiation. Its literally lower than the natural background radiation at sites in India and Brazil.
