I'd classify it as somewhere between a mistake and intentional (in order to make him look more authoritative, and thus increase clicks/pageviews). According to his IMDb profile, he does have an executive producer credit for the Batwoman's "Crisis" crossover episode, so calling him a "Batwoman executive producer" might be technically true, but is most likely misleading.
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Regarding Rose's departure, I'm curious about the timing of the news, a week after CW's upfronts announcement. I suspect exit negotiations had been ongoing for at least a little while. Plus, imagine if this had happened during a normal year, with actual in-person upfront presentations. Would news about Rose's departure had come out before CW's upfront presentation? Or, would she just have been conspicuously absent, leading to a week or so of speculation and leaks?
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I suspect that there's a very high probability the line "I'm Batwoman" will be used in the season premiere's cold open.