Does performing supererogatory prayers between Maghrib and 'Isha count as part of Qiyam al-Layl (night vigil prayer)? And does reading a hundred or a thousand verses during this time make the worshipper among the Qaniteen (devout) or Muqantireen (those rewarded with a Qintar), based on the mentioned Hadiths?

What the questioner mentioned regarding the authenticity of the hadith (definition) is correct. Al-Albani mentioned both narrations in Sahih (definition) and Da'if Abi Dawud from Qatadah from Anas ibn Malik concerning the interpretation of the verse: {Their sides forsake their beds; they call upon their Lord in fear and hope, and from what We have bestowed upon them they spend} [As-Sajdah: 16], stating that they used to wake up between Maghrib and Isha to pray. This prayer has great virtue and is considered a reason for the forgiveness of sins, and it is called Salat al-Ghaflah (Prayer of Neglect) or Salat al-Awwabin (Prayer of the Oft-Returning). Whoever performs this prayer is hoped to attain the reward of Qiyam (definition) al-Layl (night vigil) and to be among the "Muqantirin" (those who have accumulated immense reward), by Allah's permission, if he prays a thousand verses between the two 'Ishas, because the night begins from Maghrib until the appearance of the second dawn.