or plaintiff and defendant to the third-party proceedings.

It is the defenant to the pirncipal action who joins in a third-party. Hence, the name of the parties.

Garantie is shorthand for appel en garantie and has nothing to do with a warranty or insurance cover, even in this context.

The defendant joins in the third-party for 'a contribution and indemnity' to the defendant if ordered to pay a court award. Not strictly a guarantee given to the plaintiff. See my 2nd proZ discussion posting below to assignation en garantie.

The ProZ incident answer, whilst awarded 4 points, is totally misguided.

third-party plaintiff definition:
A defendant who files a pleading attempting to bring a third party into the lawsuit.

NB it is the plaintiff and not the defendant who adds a co-defendant who can also seek a contribution from the principal defendant. This is a trick Brit. Comm. etc. law finals litigation exam question. 99% of exam candidates fall into the trap and mix up a co-defendant added with a third-party joined in.