I am going to keep aside the kind of attitude problems that Anderson may or may not have. I am almost completely certain that Dhoni would not have taken this action unless there was something absolutely real.
He is far too much of a reserved person for that, and would much rather have peace in his life. Looking at other pieces of news, it seems fairly certain that his advances on this matter have been despite of the efforts of BCCI and ECB, whose default position would have been to try and not escalate the situation.
He wasn't willing to speak about it in the press room either until it was pointed out to him that Cook had basically called him a ****, or at the very least a calculating liar.
He was there, and saw what happened. No captain ever tries to get a player banned to win a series ffs; it just doesn't happen. His job, or place in the side, or his legacy and standing back home are not affected by this series.
I also don't think he would have taken this action if he felt that Jadeja was equally at fault. He has already said that he was very glad that Jadeja had not responded, and had dealt with the situation in the correct manner. He specified that, in the past, Indian players have been taken to task for responding to such incidents. So, I think he knows what he is talking about
There is absolutely no reason for him to be so steadfast on this, unless he has adopted Jadeja as a son. And people saying "If there was a push, it was just a push, and they should resolve it like men behind the scenes", I agree with your view. However, it is just like football, where a defender can get someone black and blue while playing but still not get a yellow card, and then get a red for the lightest headbutt. Also, for the scene to be resolved behind the scenes,apologies need to take place. And considering England seem to think Jadeja was as responsible, this wouldn't have materialized.
I don't think anyone should be banned for pushing when they are allowed to take somebody's head off on field with a bouncer, but the law does. Go figure.