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I personally go with the screwjob because many people see Bret as the biggest "face" of the New Generation era. Once he left, that was the end of the NG era top guys being in the main event with the exception of Shawn, Taker and Owen, who of course maintained a presence through the Attitude era (other NG era top guys like Hall, Nash, Luger and Yokozuna already left)
A lot of people have trouble categorizing Bret as being any part part of the Attitude era because he was such a major presence through the entire NG era, and because unlike Taker, Shawn and Owen he had zero presence in the company after Austin officially established himself as the new top star
A lot of people also seem to not consider Austin's career before his Vince feud (feuding with Bret, Owen, the Nation, etc.) as being part of the Attitude era, since Austin's top star status really kicked off when Austin stopped focusing on the IC title and went for the WWE title right after the screwjob
So it's probably best to use Bret's exit as the era's official starting point, even though a lot of very important Attitude era "seeds" were being planted in the company throughout all of 1996 and most of 1997