I don't think the intention with that scene was to show that it was the "start" of the Mirror universe... rather just that Cochrane was already "evil" and this was just the "mirror" version of an event we were already familiar with. The implication from the opening montage is that the Mirror universe existed long before that opening scene.
Imagine it... a million different mirror universes... all with a million different non-mirror Kirks crossing over. The conceit of these parallel universes in Trek is that similar things generally happen in them, so why not this, too?
And actually, they *did* have a cloak in the MU in "Crossover", but didn't by "The Emperor's New Cloak". So, retcon? Or slightly different MU?
Do you recall if this was definitely something you read online? Or is it possible it's from one of those "Best of Trek" compilations, or something like that?