Strike Introductions don't count properly. Nothing says that only the Daily Heroic counts toward Strike Introductions. Vanguard Srikes, Weekly Heroic and Nightfall do not count. The other one is Patrols Introductions. You can only up your score by completing the Locate Target patrols, which are far and few between. There is an indication that only VIP kills matter, not nothing explicit (I may be wrong...).