The strikes from any strike playlist do not count for the card.
Edit:
[quote]Need Strikes?
Nightfall, Weeklies, and playlists don't count towards your strike completions. The only strikes that count are the original strikes on the planets. If you go into a strike by yourself, it will only count as one. However, I completed them with one person in my fire team, and it counted as THREE every time! Find a buddy and get them done!
https://www.bungie.net/en/Forum/Post/101097919/0/0
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Dude my grimoire is 2775 I know what strikes I need to run.. my grimoire stopped increasing properly after the update