This example contradicts the rule immediate preceding it:
You use tiers (masculine) or tierce (feminine) instead of troisième when expressing a fraction, a portion of something (= one third of), as opposed to a rank or an order (= third out of).
A third party follows the some nomenclature as the third person. It is not a third of a party as if someone was having their legs chopped off. A third party to an agreement is anyone who isn't one of the signatories (ie., 'you and me'.)
It seems therefore to be an exception to the rule stated, rather than an example of the rule, as it is presented.
addendum;
This also appears to apply to "third-world", as there is a first world (NATO/OTAN countries), second world (communist bloc countries), and third-world (non-aligned countries).
The rule given in this lesson also seems to conflict with Cécile's comment on https://french.kwiziq.com/questions/view/in-the-example-j-ai-bu-un-tiers-de-la-bouteille-the-audio-speaker-is-a-woman-and-the-word-bottle-is where she says: "The adjective 'tiers/tierce' will only be used in certain expressions...It has very limited use...'.
I interpret this as 'the adjective "tiers/teirce" is only used in special cases, which you will just have to learn'.