Ad hoc arbitration is a sub set of private arbitration - that is, ad hoc arbitration is not a form of court annexed arbitration.
Most private arbitrations occur because the parties to contracts have agreed to resolve their disputes according to the rules of arbitration service providers
Sometimes contracts provide (in writing) for resolution of disputes by private arbitration but without providing any rules, or, sometimes parties have a dispute, either under an oral contract, or under a written contract that does not contain a requirement for arbitration, but want to resolve their dispute by arbitration rather than in court. This can be because of cost, the need for speed, or because of a desire for privacy. In either case, the parties can agree to arbitration without the rules of an arbitration service provider. If the parties can agree on an arbitrator (or where one is appointed by a court because of a naked requirement for arbitration), the arbitrator manages the arbitration using neutral rules set by the arbitrator himself, or agreed to by the parties and the arbitrator as part of the process of selecting the arbitrator and setting up the arbitration. This is ad hoc arbitration.
It is commonly said that reaching agreement to arbitrate after a dispute has arisen is difficult, but this really depends on whether the mutual needs of the parties to the dispute coincide with the benefits of arbitration, such as having a decision maker with industry experience, privacy, speed, and so on. Sometimes there are benefits to entering into an agreement to arbitrate after the dispute has arisen because arbitration can be conducted under rules tailored to the dispute rather than under what may have been a 'one size fits all' set of rules typically included in a pre-dispute arbitration clause.
Even where the parties can agree to arbitrate but not on anything else, it is not as if the arbitration is conducted without rules. Even without the often complicated rules of an arbitration service provider, both the federal government and the State of Oregon have passed laws
which set forth minimum standards for the conduct of a fair and impartial arbitration, and arbitrations conducted without arbitration service provider rules may invite the possibility of more supervision from a local court (if requested by a party).