That's a little bit of a Catch-22. "Engagement" is not a well-defined metric, and in practice when people try to measure it, they end up justifying the use of a

That's a little bit of a Catch-22. "Engagement" is not a well-defined metric, and in practice when people try to measure it, they end up justifying the use of any number they happen to be able to measure at the moment. So almost any number can plausibly be called an "engagement score" or something similarly vague. Your goal should be for something more specific, and ideally at least a tad more concrete. Things like "product satisfaction," "positive impression of brand," and "purchase intent" could all be synonyms for "engagement" but they are not synonyms with each other because they are all more specific and narrowly-defined. And furthermore, they're better guidelines because particular features or activities could be well-suited for one and not another, so optimizing towards one of those actually provides some focus.