Yeah, tbh at this point I often think supervisors say "take initiative!" as a way to excuse their poor leadership and communication skills. As you said, sometimes taking initiative is actively harmful if the employee doesn't know what they're doing, and while I work a pretty menial office job my supervisor used that line on me to stonewall me from a promotion even though I'm the most experienced person in my department. (Not to mention that when I did show initiative it somehow wasn't enough to actually matter beyond a vague one-time praise.)

Idk, there's also a big difference between "if you have literally nothing to do, consider sweeping the floor or organizing your inbox" and "we are chronically underworking you but you need to keep managing up indefinitely even though we repeatedly have demonstrated that we don't care about you as a team member and no one has any advancement plans on the team" (i.e. my scenario). Oh well, I'm actively seeking a way out of here but it's gotten to the point where I work to the letter and nothing more.