Tirmidhi

1:
The evidence of those people for the chapter is this very hadith, but deducing from this hadith depends on two points: First, that the wording in the hadith should be ((قَاءَ فَتَوَضَّأَ)) “He vomited, so he performed ablution,” whereas this wording is not preserved. In most sources of hadith, in the narrations of most narrators, the wording is ((قَاءَ فَأَفطَرَ)) “He vomited, so he broke his fast,” and even after ((فَأَفطَرَ)), the wording ((فَتَوَضَّأَ)) is not present, or it is as it is in this narration, that is, ((قَاءَ فَأَفطَرَ فَتَوَضَّأَ)) “He vomited, so he broke his fast, and then performed ablution.” And from this wording, the obligation of ablution is not established, because it happens that after vomiting, a person becomes weak, so he breaks his fast and, for the sake of cleanliness, performs ablution. And the Messenger of Allah (sallallahu alayhi wa sallam) was even more fond of cleanliness. Moreover, this was merely an action of the Prophet (sallallahu alayhi wa sallam) with which there is no command from him. Second, even if the wording ((قَاءَ فَتَوَضَّأَ)) is preserved, the “fa” in ((فَتَوَضَّأَ)) could be for causation, meaning: “He vomited, so he performed ablution because of it,” but this is not definite; rather, this “fa” could also be for sequence, meaning: “He vomited, and then after that performed ablution.”

2:
The evidence of those people is the narration of Jabir (radi Allahu anhu), which Imam Bukhari has mentioned in suspended form (ta‘liq), that the Prophet (sallallahu alayhi wa sallam) was in the expedition of Dhat ar-Riqa‘ when an arrow struck a man and blood began to flow, but he continued his prayer and kept performing bowing and prostration in that state. It is evident that the Prophet (sallallahu alayhi wa sallam) certainly knew of his prayer, because this prayer was performed while standing guard, which the Prophet (sallallahu alayhi wa sallam) had commanded him to do. Despite this, the Prophet (sallallahu alayhi wa sallam) did not command him to perform ablution or to repeat the prayer.

Benefits and Issues:
If one vomits deliberately, the fast is broken and making up (qada) becomes obligatory, as opposed to when vomiting occurs involuntarily.
If vomiting occurs on its own, neither is the fast broken nor does making up (qada) become obligatory.