Branches of Orthodox Judaism
"Traditional" can be either a synonym for Orthodox (especially when using the dichotomy Traditional/Liberal), or a synonym for "Conservadox." In Israel, "Traditional" is equated with the MASORTI.
"Centrist/modern/cosmopolitan" (colloquially [sometimes pejorative, sometimes affectionate] kipa sruga [crocheted skull cap]). Adherents usually approve many aspects of secular culture, especially secular education, in addition to traditional Torah study. They tend to be Zionists. The precise term depends on the speaker-Rabbi Norman Lamm uses "centrist," Rabbi Shlomo Riskin uses "cosmopolitan" and Rabbi Emmanuel Rackman uses "modern." The Union of Orthodox Jewish Congregations, Yeshiva University, and the Rabbinical Council of America in some sense represent this group. In Israel, the Mizrachi organization is a wellknown representative.
"Yeshivish" (colloquially, [sometimes pejorative, sometimes affectionate] "black hat" or "black") suggests an Orthodox outlook in which the focus of life is Torah study, as is done in Lithuanianstyle yeshivot. Secular culture is either tolerated or criticized for its corrupting influences. This group tends to be "nonZionist" in the sense that they love the land of Israel and its holiness (many spend years in Israel for Torah study), but are unenthusiastic about secular Zionism and Israeli secular culture. Agudath Israel tends to represent this group.
In Israel, the "Dati/Chareidi" distinction is more a matter of attitude toward Zionism than of political affiliation or religious views. The Dati tend to be more supportive of Zionism, with the Chareidi not having much belief in the modern Jewish state. Please note that these are general positions; individual members may hold different views.