Is a Muslim's saying, "O Allah, save me, and I will be among those who remember You," or "O Allah, save me, and I will be among those who are grateful to You," considered a vow that must be fulfilled? And is performing the five daily prayers sufficient to fulfill it? And what should one who said it do to avoid it being a vow?

A mere verbal promise, such as saying "I will be among those who remember You," does not constitute a valid vow (nadhr), because fulfilling it is not obligatory. Even if it were assumed to be a vow, it could be fulfilled by performing ten verses (rak'ahs) at night, or by reciting ten verses; as the five daily prayers are not usually intended by such a phrase. As for an exception, its validity is conditional upon intention and connection to the wording of the vow. If there was no intention for a vow, then the default is acquittal of responsibility. However, if there was an intention for a vow, it can be fulfilled by expressing gratitude and praise to God. Indeed, Noah (peace be upon him) was very grateful. One should be wary of vows of recompense.