Obviously inaccurate language affects trust in the police and likelihood to follow police instructions.
To paraphrase Sal Grover, if you lie about obvious things, I'm not going to trust anything you say..
It affects whether or not people believe the police are doing their job and will act in everyone's best interests. And therefore whether people will cooperate with police or even engage with them at all.
It's clearly not in the interests of any missing person for the police to make things confusing about who they are and who 'members of the public' should be looking for.
If they're 'asking for help' gaslighting the people who might or might not help doesn't seem the best approach.