Just to clarify on this point - if the foul is the push then it doesn't matter if he was trying to kick the ball at the same time. The foul is the push - and that one doesn't offer the option of a yellow card.
Look at the way it's written. Whether it's a genuine attempt to play the ball is a whole separate consideration to whether it's a hold, pull or push.
If the offence is one of those three, then it simply doesn't matter whether it was an attempt on the ball or not. It's red.
Now, often a player will commit multiple fouls at once. Push and a trip for instance. In that scenario, the referee has the flexibility to do what he sees best - and he can skew that scenario to match a YC if he sees fit.
Impossible to tell what happened on the video. The manner of the fall is what suggests'push' to me - and if that's the case, then it's DOGSO. But it might be a trip - and if so, then a caution is fine, if it was a genuine attempt on the ball.
As for other DOGSO criteria? well, defenders isn't an issue - only other defender is too far away, given the striker is already within shooting range or within a few yards of it - the only other defender is further upfield than the player. There might be a case for direction being unmet if the last attacking touch was heavy and angled away from goal.