I think you're referring to Ian Walker's study. Yes, motorists on average passed helmeted cyclists closer than non-helmeted ones. The difference in average passing distance wasn't huge (1.3 m vs. 1.2 m) and was still a safe distance.
Does it? That's a supposition, not a scientific fact. Presumably they all paid some attention by passing at a safe distance. I don't see why someone paying more attention would pass further away.
I have not read the study, but I would hope that they had the cyclists wearing the same clothes and riding on the same stretch of road in all tests. Otherwise you will end up with the situation where helmeted riders are more likely to be riding fast on roads that aren't as safe for cycling, Which would naturally lead to differences in driver behavior around them.
Even better if you are a woman in summer dress. Then the drivers give you way more space and are super careful around you. It was researched.