Don't really see why motive matters in that case. The idea of misstating the problem doesn't exist (the bonus and the cost reductions are intrinsically tied to one another) and the additional clarity doesn't guarantee the desired behavior. It's possible a subordinate might not help you if they don't feel your personal gain is beneficial to them.

If a manager says "I need to reduce costs by 10%", the specific desired business outcome is clear. Reminding your subordinates that your pay and continued employment is tied to you completely the tasks and requests of the people you answer to doesn't seem needed. This isn't about "some organizations", it's just about efficiency.

Organizations always want to reduce costs, there is always pressure to be efficient and reduce costs.

Usually the current costs are the cost of business, and if there is an externality ie economic downturn, some crisis, big lost deals, then there may be renewed pressure at which point this has already been communicated by the leadership to the org and everyone is on the same page.

Trying to reduce costs out of the blue is not usual, it suggests some individual thinks there is some inefficiency after studying the issue, or its a management consultant recently hired to explore that, but usually the people being requested for information would be briefed and know exactly the reason and why.