His relative, R' Yaakov Lifschitz, wrote a sefer called Toldos Yitzchak where he writes in the introduction (at the bottom) that R' Yehuda Lifschitz helped support his efforts. That sefer is about the life of R' Yitzchak Elchanan Spektor*, the namesake of the Rabbi Isaac Elchanan Theological Seminary, a part of Yeshiva University.
R' Yehuda Lifschitz also wrote a Sefer called דור ישרים, published posthumously, as a response to and condemnation of דור דור ודורשיו by Issac Hirsh Weiss. It has approbations from several people, primarily from the Litvish world, such as R. Chaim Ozer Grodzinski, and figures from Volozhin, Telz and other places. As the introduction in Derech Emuna lays out, it was also written in response to the same work.
He also wrote a sefer called נחל יהודה which seems to be more conventional Torah ideas rather than refutations of other ideas. It doesn't contain much introduction.
It seems quite likely the first entry here is him, but for an additional middle name that is not printed in the seforim. But the timeline and father's name match. That would mean he lived from 1829 to 1905.
So from all of this his general sphere of influence was the traditional Litvish groups of the 19th century.
*As far as I can tell, that is the intended person, I didn't see the last name mentioned.