Brother Cox strongly emphasized serving "this way" (with an outward-moving gesture). This, to me, seemed to define the entirety of his lecture. In all of his examples--George Washington, Alvin York, Ghandi, etc.--the greatest thing about them was that they worked to bring the best for other people because they loved them.
Brother Cox laid out the central dogma, so to speak, of servant-leadership: desire, love, serve. I loved how this corresponded to the message in my selected book, The Servant (cool, huh?): leadership is built on authority is built on service/sacrifice is built on love is built on will. True success as a servant-leader comes when one wills or desires for others to have, to do, and to become their best. As someone that Brother Cox quoted said, "The genesis of servant leadership is building others up."
Building and lifting others is what Heavenly Father is all about. He says, "For behold, this is my work and my glory--to bring to pass the immortality and eternal life of man" (Moses 1:39). We, as covenant members of Christ's church, have pledged to "press forward" in our lives "with a love of God and of all men" (2 Nephi 31:20). We show our love for God by keeping his commandments (John 14:15) and by helping His other children to do the same by serving them, meeting their needs so that they can grow and progress (which is just exactly what Jesus does; see John 13:34).
A small example of application: in my role as a Sunday School teacher, I try to help my class members fulfill their need to understand the doctrines of the gospel so that they can grow in understanding and in love for their Father in Heaven.
This, then, is servant leadership: to love those in your sphere as children of God; to show them the goal, the Light of the world; and then to give your all (time, talents, energy, resources) to boost them on their journey upward.