Cards are the visual elements that display inside of Tweets. Broadly, there are two types of cards:
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Dev cards that are configured using markup inside web pages, and scraped by Twitterbot. These show up when a link to a page that contains the markup is shared in a Tweet.
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Ads cards that are configured via ads.twitter.com and that show up in promoted Tweets.
There is no card_count property shared in the standard API. I am not clear what this value would be for. The Engagements data is available in the Engagements API, and does not refer to cards.