To what extent is the topic that includes the Hadith: "He who recites ten verses will not be recorded among the negligent; he who recites one hundred verses will be recorded among the devout; and he who recites one thousand verses will be recorded among those who amass great rewards," sound, along with its explanation that links night prayer to "your name by night" based on the number of verses recited, and concludes with poetic verses?

The aforementioned hadith (definition) is authentic, and what was mentioned in the question regarding admonitions and advice as an explanation for it falls under its general meaning. Every servant has a characteristic by which they can be titled or called, so they can be among those who remember Allah often or those who are grateful. There are many hadiths with this meaning, such as what is narrated in Musnad al-Harith: "If a man wakes up at night and prays two rak'ahs, he will be written among those men and women who remember Allah often."