What's a more formal name for a "third" party?

A mediator helps resolve disputes between conflicting parties.

Besides mediator, "One who negotiates between parties seeking mutual agreement", consider negotiator, moderator and arbitrator, "A person to whom the authority to settle or judge a dispute is delegated."

I think your proposed "neutral party" is accurate and appropriate. I don't think there should be any concern about it being informal. If Switzerland calls it "neutrality", it should be good enough for you.

All those suggestions are valid and good but those would not be third parties, any more than a judge or jury is a third party.

So your dilemma would be resolved by reference to those nominal titles while refraining from any reference to a third party, if in fact there is no involved party who is also not aligned with party 1 or party 2.

If you are expressing in a legal sense, in a mediation (non-binding) or arbitration (binding) setting, then the enumeration of parties (between party the first and party the second) would depend on the nature of the dispute, hence of the action. An action in contracts would typically pit a first party payor against a second party payee. In torts a first party plaintiff against a second party defendant, contrariwise with countersuit, In fictional literature, you might say that party 1 is the protagonist, Party 2 the foil, party 3 the antagonist. In grammar, party one is I, two is you, three is (s)he.

When you say, to the effect, neutral, unbiased, disinterested, no finger in the pie, you answer your own question: such a person can be neither first, second nor a third party.

The situation would be called as triangulation or in some context Back-channeling (if to some extent secretive). So following would be possibly suitable as formal.

Triangulating intermediator or Triangulating arbitrator or Triangulating intercessor

Triangulation / Back channeling