Is the Witr the same as Qiyam al-Layl, and does it have a specific number of rak'ahs, and is it performed after finishing Qiyam al-Layl?

There is no contradiction between the intention for Witr and for Qiyam al-Layl (night prayer), as Witr is part of Qiyam al-Layl. The worshiper can intend for the rak'ahs preceding Witr to be Qiyam al-Layl, and that Witr consists of the final odd-numbered rak'ahs (one, three, five, seven, nine, or eleven). There is no maximum limit for night prayer; the more a servant performs, the greater their reward.

A Muslim can pray as many voluntary rak'ahs as they wish during the night, then conclude their prayer with Witr in one of the legislated manners. What precedes Witr is considered part of Qiyam al-Layl. The Sunnah and scholars have differentiated between night prayer and Witr in terms of ruling and manner. Witr's obligation is disputed, whereas night prayer is recommended.

As for the manner, night prayer is performed two rak'ahs at a time, whereas if Witr is five or seven rak'ahs, one only sits at the very end. If it is nine rak'ahs, one sits after the eighth, recites the tashahhud, then stands for the ninth rak'ah and performs the salutation. Witr should not exceed eleven rak'ahs, while the door to absolute voluntary prayers is wide open.