Your vision is your description of the place or condition to which you've been called by God. Casting vision is inviting people to the place or condition to which you're heading, and where you will hold space for the people you're inviting to join you.
In the same way that a good leader responds better to praise and support for their team or vision than they respond to praise for the leader, because a good leader has already paid such a price that their own ego needs have been relegated to a secondary position, casting vision focuses on the destination or the space the leader is holding, rather than engaging people in the head trash, tactical considerations, and temporary objections or distractions that exist between where they are today and where the vision lives.
In Scripture, we see Paul say, "follow me as I follow Christ," or we see Jesus say that He is the way to the Father. Leadership is best measured by the fellowship it facilitates between the follower and the Father. Visioncasting is an usher's sweeping hand gesture to the correct seat in the theater, where they will encounter the show – the purpose of the gathering in that space. It is a poor usher indeed who thinks the usher is the show, or that the distractions of the theater-goer are more compelling than the show for which the audience is gathering. The usher who falls into these traps of self-expression or distractions becomes an enemy to the audience, the theater-goer/follower, and certainly to the ones for whom the stage has been prepared.
Why the elevator image? Because vision is about a specific destination, not about the process. At least not for kings (priests and peers can work differently because they have different roles and different obligations). The elevator asks, "Which floor? Push the button when you're ready and I'll take you where you want to go." The elevator holds a "vision" for each floor as a destination, and has built processes for moving people to those destinations, but elevators don't help people process through their fears of elevators, heights, claustrophobia, sneezes, farts, or weirdos. That's someone else's job (preferably the person who wants to get somewhere).
There is one exception to this, and even it has limitations. We must speak vision for our children. Their image of their parents is so deeply interwoven with their longing for their Heavenly Father, and parents are the first exemplars of defining love and direction. But a parent who does not strive to place their child's hand into the hand of The Father will ultimately create "deconstruction" issues when the child and God begin to reach for each other directly. We have a craving for our Father ... we only develop mommy and daddy issues when our parents fail to cast vision and make the true introductions as their children truly require.