No, I don't think you're right. If the sentence was "Det började snöa igen", then you would be right, but the sentence has "har" in it
Det har borjat = it has started or it's started.
Det började = It started
NOTE: You wrote that it corrected to Its started snowing again.". There is an error in that sentence. It should be It's started snowing again. There should be an apostrophe--which in this case would show that "It's" is a contraction (of either "It is" or "It has".) "Its" shows possession, as in "The dog bares its teeth.)