Hi Borek,
The early suttas don’t explain this any further, and there has been some debate on what it actually means, with no firm consensus. The Theravada commentary tradition regards the “way” or “path” as a single mind-moment, but most modern scholars disagree with this. They don’t have a universally accepted interpretation either, however.
I think of it like this:
The insights into a specific stage of the path don’t happen in a single moment. It can take some time for them to assemble in the mind. For example, when the Buddha achieved his awakening, tradition says he did it in three stages of the night; see MN36.
I understand the period of attaining these insights to be the Buddha’s “way to arahata”. During that time he was a specific type of noble person, one who was going to attain full awakening very soon. He wasn’t really a non-returner, because he wasn’t going to reborn in a non-returner realm anymore; but he also wasn’t a full arahant yet, because he still had to have the final insight. So therefore the special status as “on the way to arahata”.
(Leaving aside the issue that the Buddha may have jumped straight to full enlightenment here. His one-night period of getting awakened just illustrates the basic idea.)
So not all anagamis are “on the way to arahata”, in this technical sense. It’s only those whose awakening is very close, those who are breaking through to some of the final insights.
It’s similar for those on the way to stream entry. It can take a few days, even, for the mind to gather all the insights, until true stream entry occurs. Between the initial, irreversible moment that triggers these insight on the one end, and the final complete insight into Dependent Arising on the other end, that’s when they were “on the way” to stream entry. After that, they are full stream enterers. Before that, they were ordinary persons (puthujanna). In between they are neither ordinary persons, nor real stream enterers. They are “those on the path to stream entry”.
As I said, there isn’t too much in the suttas to support this either way. But thinking of it like this, would fit for example the idea that someone on the way to stream entry supposedly will attain stream entry before they pass away. This doesn’t mean they are magically kept alive for years while on the way to stream entry.
It instead means they are so close to full stream entry that if they die, they will gain the final insights into Dependent Arising before moving on to a next life.
That’s how I see it. Sorry there isn’t much more definitively to say based on the early texts themselves.
Good luck with your book!