Modern Orthodoxy has its roots in R' Hirsch's "Neo-Orthodoxy", which was defined by Torah im Derech Eretz. Modern Orthodoxy is actually defined by R' Joseph B. Soloveitchik, and the slogan, "Torah U'Madah". (Although R' Soloveitchik did not actually create this label in his writings, nor did he coin the term.)
There are three works of R' Soloveitchik which define Modern Orthodoxy: Halakhic Man, The Lonely Man of Faith, and The Halakhic Mind. In Halakhic Man, R' Soloveitchik expresses the 4 amot of halacha, and how it is very down to earth, this world endeavor. A sort of spirituality through legalism. In Lonely Man of Faith, R' Soloveitchik compares the "two stories" of Bereshit, comparing and contrasting Adam I and Adam II, and explains how we need to emulate both. Adam I and Adam II can be seen as a parable for Torah (Adam II) and Mada (Adam I). Lastly in The Halakhic Mind, R' Soloveitchik describes the historical relationship between strict science and philosophy, and how that same relationship can be applied to halacha.
The phrase "Torah U'Madah" generally captures the definition of Modern Orthodoxy. Where secular studies, and world sciences, (i.e. G-d's creation) becomes a spiritual uplifting influence on the harsh legalism that is pure Halacha.
The other culturally defining aspects of Modern Orthodoxy such as Zionism, and women studying Talmud are outgrowths of the central core, which is "Torah U'Madah".
R' Joseph B. Soloveitchik was the "Rosh Yeshiva" of YU (Yeshiva University). YU, as a school, has the slogan of "Torah U'Madah" and with that phrase they have defined Modern Orthodoxy. The phrase was first coined by Bernard Revel, president of Yeshiva College, which is the undergraduate school of Yeshiva University. Before R' Soloveitchik, the precise meaning of Torah U'Madah was unclear, but with his teachings, it gained its definition.
R' Lamm, president of YU, once rebranded Modern Orthodoxy as Centrist Orthodoxy. But he later regretted it, as he was quoted in 2010:
I quickly saw that it was totally misunderstood. “Centrist” does not
mean that you have Conservative and Reform Judaism on one side and
“real Judaism” on the other and we are somewhere in the center. That
is nonsense. Rather, it means that we are the center within the
Orthodoxy community. I now try very much to discourage the use of the
word “Centrist,” because it has been misunderstood.
And he later explained:
Some sociologists distinguish between “Modern” and “Centrist”
Orthodoxy – which is narishkayt (foolishness). Of course there are
varieties within Modern Orthodoxy, just as there are varieties within
Charedi Judaism; none of us is monolithic. But there is absolutely no
essential difference between these titles in terms of the group they
However, today because of Open "Orthodoxy", who also claim to be Modern Orthodox, yet do not come from YU, and are more social justice oriented, YU and Lamm prefer the term Centrist Orthodox.
The Rav and his teachings however, are still the main defining element of Modern Orthodoxy, despite the fact that many people do not live up to the standard.