1. An application to compel a solicitor to deliver up monies is styled "In the matter of A.B., a solicitor And in the matter of an application pursuant to the Courts inherent jurisdiction over its own officers of court". You can do it yourself but you might be better off getting a solicitor and barrister to do it for you. The application will be based upon an affidavit in which you set out the facts of how the monies came to be in the solicitor's hands. You will recount the joint receipt, the giving of the monies, the terms upon which it was to be held, its transmission to another solicitor and that you are seeking it back.
2. This application will be made in the High Court. Its aim is to get back your monies summarily. Here you should really be bringing it agains the second solicitor who now has the monies in his hands.
I need to know the Statute in NI under which the Summary Motion by Summond and Affidavit is brought?
Is it Specific Performance of my contract?
I can't rate this until I know
3. There is no statute. As I stated in my earlier answer, it is "pursuant to the Courts inherent jurisdiction over its own officers of the court". This is judge made law over centuries not statute law. It is very different to specific performance, as there are no procedural bars to the remedy to get the solicitor to pay over the money.