Let me try to explain what I mean:
For as long as I can remember I have agreed with the classic definitions of the Priesthood as "the power and authority of God" and "the authority to act in the name of God" - but I have disagreed that such a definition is limited to men who have been ordained to offices within the administrative Priesthood structure. I have believed that there is a huge difference between "the comprehensive power and authority of God" and the implementation of organizational Priesthood authority.
I believe ALL who covenant to take the name of God upon them (inside or outside the Church, irrespective of sex or age and no matter what specific name they use) have the authority to act in His name in some way and to some degree (unless they lose it through the type of abuse described in Doctrine & Covenants 121) and all who have access to the Holy Ghost can exercise His power (to some degree). In fact, I believe that is inherent in the very words of the sacramental prayer, the way that the Priesthood is exercised in the temple and our beliefs concerning revelation that can be received by all.
Given this fundamental belief, I define "the formal, organizational Priesthood" as it currently is implemented in the LDS Church in administrative terms - since that is how our own Article of Faith presents it. Therefore, I would tweak the definition of that Priesthood as follows:
"the authority to administer (oversee the performance of) eternal ordinances and to administer (oversee the performance of) the activities related to those ordinances, including preaching the Gospel"
I'm not arguing against the current structure of the organizational Priesthood responsibilities; not at all. I'm just saying that there is a difference between the fullest definition of the power and authority of God and how the Priesthood is exercised currently within the Church - and I'm OK with that.
Also, I'm not sure that everyone will understand what I am saying in this post (and I am concerned a bit that some will misunderstand), but I really don't think what I'm saying is controversial in any real way. The definition above allows for MANY actions to be performed without organizational Priesthood authority but still through the more general authority of God - and, I believe, it is much truer to the actual way the Priesthood is used in the Church than the more common definition. I believe this is part of what Elder Bednar was saying in his talk about light and revelation, even though he didn't address the specific application I am making in this post.
[Also, in the interest of full disclosure, my belief that women who have been endowed in the temple by the administrative Priesthood leave the temple with real power and authority as a result of that endowment also influences how I have tweaked the organizational, formal definition to focus more narrowly on "administrative rights" (as assigned responsibilities) rather than the more comprehensive "power and authority of God".]