The term “Dati” is used to mean what would be called “Orthodox” in the US – which runs the gamut from barely Shomer Shabbos to Satmar. There are subdivisions is Israel, as there are in the US – including “Dati Lite”, which would be the very left wing of the spectrum, and “Chareidi”, which is NOT the same as American Yeshivish, though the “uniform” of black hat/black suit/white shirt is the same (American Yeshivish is generally much more open to secular/professional education and working with the outside world than Israeli Chareidi). That’s the reason many American Yeshivish Olim have issues, particularly with their children, when they move to Israel – they think they are Chareidi and send their children to Chareidi schools, but find that the Hashkafa of the school is not the same as the Hashkafa in the home, leading to confusion and, at times, rebellion.
In Israel, you also have “Torani”, or “Dati Le’umi Torani”, which are the very Frum Dati Le’umi – who can be just as Makpid on Shmiras Torah u’Mitzvos as the Frumest Chareidi, but have a different Hashkafa – they tend to view the existence of the State of Israel, or at a minimum, the widespread return of Jews to historic Eretz Yisrael, as having religious significance, and act accordingly. That’s part of the reason you don’t really have this group in Chu”l – for them, living in Israel has such religious significance, they would not consider living outside of it.
There are, of course, those who are on the borders of various groups – Chardal, or Chareidi Le’umi, who are somewhat between Chareidi and Dati Le’umi, though there can be significant overlap between this group and Torani. And, as we all know – two Jews, three opinions – so all feel free to create their own unique “group” that they alone to belong to…
an Israeli Yid