Engaged spirituality is any spiritual practice that embraces work in the world to improve social conditions and the well-being of human and natural communties.
Engaged evolutionary spirituality arises from one's realization that one is a facet of the evolutionary process. One's spiritual practice of evolution, then, involves playing one's evolutionary role as consciously and co-creatively as possible. To the extent one's practice and the evolutionary process become one motion of Spirit, one is practicing engaged evolutionary spirituality.
The term "engaged spirituality" was inspired by the term "engaged Buddhism", originally coined (according to Wikipedia) by Vietnamese Zen Buddhist teacher Thich Nhat Hanh. During the Vietnam War, he and his sangha (spiritual community) made efforts to respond to the suffering they saw around them. They saw this work as part of their meditation and mindfulness practice, not apart from it. Since then, the term continues to apply to Buddhists who are seeking ways to apply the insights from meditation practice and dharma teachings to situations of social, political, and economic suffering and injustice.
A difference between most engaged spiritual practices and engaged evolutionary spirituality is that the former usually focus on relieving suffering and injustice, while the latter focus on tapping existing energies -- including suffering, injustice, and catastrophe, as well as individual passions, collective yearnings, and systemic trends -- to stimulate individual and especially collective evolution in positive directions.