When is a statement considered a legitimate vow (nathr shar'i), and is a specific wording or utterance by the tongue required? What is the ruling on a vow contingent on a condition if it does not include "for Allah is upon me" (لله عليّ) or the word "I vowed" (نذرت)? And what are the consequences of not fulfilling it?

The wordings mentioned in the question are not considered a vow unless a vow is intended by them. In that case, the vow is concluded and must be fulfilled. Whoever is temporarily unable to fulfill the vow is not obliged to do anything until they are able to fulfill it. Whoever is permanently unable to fulfill it must offer an expiation for an oath. As for the wording, "If God wills, I will do such and such," if it is intended as an exception, then the vow is not concluded. And whoever intends to pray twenty rak'ahs if they succeed is not obliged to do anything, because a vow is not concluded by mere intention without verbal utterance.