Thanks for the question. I defined Franciscan poverty the way I did in the post you referred to because my sense of Francis' take on poverty was that he let go of anything that obscured or prevented his complete dependence upon God; the necessary corollary is that he let go of anything that prevented him from being truly himself before, with, and in God. He demanded his followers also relinquish things so that nothing would stand in the way of their relationship with God. Because God is truth itself this relationship with God is the source and ground of standing in one's own truth and being oneself. The same is true of God as love. Because God is love-in-Act one is able to be wholly oneself in God's presence; one needs no props, no other sources of Selfhood than God alone. The very essence of faith (and love of God) is the ability to stand before God as the person one is. Thus, Francis very much wanted those who followed him to stand naked (so to speak) before God, and more, to become entirely transparent to the grace (presence) of God in Christ Who is working in and through them.
Similarly, it was this latter posture which was and is at the heart of Franciscan poverty and which material poverty was/is meant to serve. I have written here before about this view of evangelical or religious poverty; my own vow is defined in these terms rather than in terms of material poverty --- not because I don't embrace material poverty but because I know that if I measure matters in terms of my dependence on God and focus on or give that priority, material poverty will largely fall into place. The opposite is not as true, at least that is how it seems to me; material poverty can foster dependence on God, but it need not do so. In any case, the two things go hand in hand so that in formation as a Franciscan, for instance, material poverty is a given and exhaustive dependence on God to be the one one is called to be is the focus of the spirituality.
As noted, this is the way I view the evangelical counsel of poverty. My vow reflects this explicitly and reads: [[I recognize and accept the radical poverty to which I am called in allowing God to be the sole source of strength and validation in my life. The poverty to which my brokenness, fragility, and weakness attest, reveal that precisely in my fragility I am given the gift of God’s grace, and in accepting my insignificance apart from God, my life acquires the infinite significance of one who knows she has been regarded by Him. I affirm that my entire life has been given to me as gift and that it is demanded of me in service, and I vow Poverty, to live this life reverently as one acknowledging both poverty and giftedness in all things, whether these reveal themselves in strength or weakness, in resiliency or fragility, in wholeness or in brokenness.]] (cf.,Everyone is called to the Evangelical Counsel of Poverty)
There is a strong dimension of the richness and meaningfulness of this kind of poverty; it is a paradoxical reality and I wanted to capture that in the vow itself. The reverent approach to life lived in this way, and to everything and everyone one encounters, was also something I needed to capture as an integral dimension of such dependence. When we can stand before God in the way Franciscan poverty calls for, we can be open to all of creation in a reverent and accepting way. In any case, though I might write a slightly different vow today (I first used this vow in 1976), the priority given to complete dependence on God to be the person I am called to be would still be it's heart. I think my vow of evangelical poverty is essentially Franciscan, but I did not consciously draw it from Franciscanism; instead it came from my experience of God's presence in my life and from reflection on the Pauline and Markan theologies of the Cross. I hope this is helpful!