Bukhari

Hadith Commentary:
In the translation of the chapter heading, rent was mentioned, while in the hadith, phlegm is mentioned.
Since both are considered human waste, the ruling for both is the same. In the mentioned hadith, there is no restriction regarding prayer.
However, further on, the same narration is coming from Adam ibn Abi Iyas, in which there is a restriction regarding prayer.
Imam Nawawi rahimahullah states that this prohibition is absolute.
That is, whether one is in prayer or not, whether in the mosque or outside the mosque, spitting in the direction of the qiblah is prohibited.
In the previous chapter, there was mention of cleaning spit with one's hand, and here, scraping it off with a pebble is mentioned.
From this, it is evident that sometimes the Prophet sallallahu alayhi wa sallam did it this way, and sometimes that way; in both manners, the objective is to keep the mosque clean.

Hadith Commentary:
Shah Waliullah Muhaddith Dehlawi rahimahullah, in his Sharh Tarajim Bukhari, writes that with this chapter heading, Imam Bukhari rahimahullah refutes those individuals who, based on the mentioned hadith, have argued that nasal mucus is impure. Their reasoning is that the Messenger of Allah sallallahu alayhi wa sallam personally performed this act not merely for the cleanliness of the mosque, but rather because he wished to purify the mosque from impurity. However, it is possible that Imam Bukhari rahimahullah’s intention was indeed to refute these individuals, because Imam Bukhari generally takes such matters into consideration in his chapter headings, and perhaps he mentioned the narration of Ibn Abbas radi Allahu anhu for this very purpose, as it serves to refute those people. But in my view, the best explanation is that when Imam Bukhari rahimahullah possesses a hadith through multiple chains of transmission, he does not mention all those chains in one place; rather, it is his habit to mention each narration under a separate heading, and, taking into account the wording of the hadith, he introduces variety in the chapter headings and titles. The real objective is to mention the different chains of transmission of the hadith, otherwise the purpose of “Bab al-Buzaq” (Chapter on Spitting) and “Bab al-Mukhat” (Chapter on Nasal Mucus) is one and the same.

Sometimes, Imam Bukhari rahimahullah’s objective is not the chapters themselves, but rather the narrations. Thus, here too, the objective is not the chapters, but the narrations.

And Allah knows best.