That isn't right, it is impossible for a player to be banned for 3 games for assault on a match official. The offences in this space are (from the FA disciplinary handbook) …
96 The three categories of offence against Match Officials are as follows:
96.1 Threatening behaviour: words or action that cause the Match Official to believe that they are being threatened;
96.2 Physical contact or attempted physical contact : examples include but are not limited to: pushing the Match Official, pulling the Match Official (or their clothing or equipment), barging or kicking the ball at a Match Official (causing no injury) and/or attempting to make physical contact with the Match Official (for example, attempting to strike, kick, butt, barge or kick the ball at the Match Official); and
96.3 Assault : acting in a manner which results in an injury to the Match Official. This includes spitting at the Match Official (whether it connects or not).
The offence mentioned by the referee undoubtedly falls under 96.2, and the sanction for that is a minimum suspension of 112 days, as below. Even if they did it under 96.1 that carries a minimum suspension of 56 days.
Physical contact or attempted physical contact:
101.3 suspension from all football activities for a period of 182 days, with a mandatory minimum of 112 days from all football and football activity.
101.4 a fine of up to £150.
At a guess I would say the 3 game ban was for his sending off that led to him pushing the referee, but he either wasn't charged for assault due to the report not being clear, or there being a hearing but the referee couldn't or wouldn't appear. I've heard of referees submitting a standard WGS send off report and putting details of extraordinary incidents in the comments field, that won't trigger a separate charge and they must raise a standalone extraordinary incident on WGS.