I’d like to try to answer your question as I am a product of this type of background.
“Modern Orthodoxy” was the orthodoxy that existed in America in the early 1900’s just by virtue of it’s existence. No one initially defined it.
Many things the old timers brought over from Europe were not continued by the next generation.
Those who remained Orthodox – kept Shabbos, but were often very unknowlegable of the details of Halacha in all areas. Ladies covering hair, carrying a key of Shabbos, correct procedures for warming food on Shabbos, making Brochos on food, Shatnez, modest clothing, etc. These things weren’t kept properly by many. This was what modern Orthodoxy was – Jews who weren’t the bearded European parents, who were educated secularly and limited in their Jewish knowlege.
In addition there were the ideological changes of the generation regarding education, knowlege and type of jobs available to Jews.
At some point Rav Soloveichik defined Modern Orthodoxy as a goal for those who were remaining Orthodox – more stringent Halacha, and the incorporation of the study secular subjects and worldliness in order to bridge the gap between the Torah and the Jews of the day.
There are those today who call themselves modern Orthodox – who go mixed swimming, etc. – this is not the Modern Orthodox the Rav wanted people to aspire to. He wanted to get people to keep the Torah properly, to keep mitzvos more correctly, but was also incorporating secular knowlege and study into it.