When people colloquially say "you cannot prove a negative" they are usually referring to the fact that absence of evidence can not be used to deduce non-existence of some phenomena ("a negative"), whereas the factual discovery of a phenomena can be used to deduce that the phenomena exists ("a positive").
They are therefore not referring to formal negation but rather making a point about deductive vs. inductive reasoning and the asymmetry of these two related questions (existence vs. nonexistence).
There is a bit of nuance to add here in that practically speaking you can't really "discover a fact" by direct observation. But again this is a colloquialism as most laypeople will accept what is directly observable under their noses as factual rather than a noisy data point of one.
Sure I can agree with that.
However, I think that is sort of a special case that's easy to resolve. It only comes up when they are already in the business of learning logical proofs & will likely be looking to learn from someone or a textbook who will most likely clear that up for them...
Chances are that person already has a baseline level of competency in logical thinking, or, if they don't, they soon will learn and are open to it. They've at least additionally already mastered the colloquial meaning of the phrase and are simply a bit overzealous with it's use (which should be reigned in as you aptly point out).
On the other hand, when people don't understand "you can't prove a negative" in social situations unrelated to formal logic, it's generally observed they are up to their eyeballs in conspiracy thinking and are so lost in magical thinking that they've abandoned even informal rule of thumb levels of logic.
Those are truly sad situations with deep (inter)personal, social, and political consequences, especially if they go on to harm others based on their misunderstandings.
Ironically it seems we both have less faith in the competence of others, albeit in different ways lol