Point of fact being "did x happen". Point of law being "did x happening violate law y".
So if we follow Menard's Ontario case, it becomes one question to whether he did in fact show his badge or do some action to make himself out to be a peace officer, and then a question of law to whether the action proven to be a fact constitute a violation of the law.
Can't remember if he was charged under 'a' or 'b'.130 (1) Everyone commits an offence who
(a) falsely represents himself to be a peace officer or a public officer; or
(b) not being a peace officer or public officer, uses a badge or article of uniform or equipment in a manner that is likely to cause persons to believe that he is a peace officer or a public officer, as the case may be.
As an interesting through, him presenting himself to the guards saying "i'm a peace officer" after they have been warned he typically does stupid stuff like that might not be a violation of 130(1)(b) because there's no means of confusion there.
However, presenting his badge to some cop in Ontario at a random stop might be a source of confusion.
Here, we don't even know what he did exactly, so its hard to match the law to the act.