Salam. Brother, I have learned that if we want to pray tahajjud, we have to get some sleep first. If we do it right after 'isha prayer before sleep, it is only called qiyamul lail. Does that not count as tahajjud, even if our niyah is to pray tahajjud?
Alaikumassalam wa rahmatullah,
Tahajjud means “to keep vigil”, to avoid sleep for the sake of something. Qiyām al-layl means “staying up at night”. Linguistically the two have the same meaning, but some scholars choose to differentiate between them, reserving the word tahajjud for interrupting one’s sleep to pray, while considering qiyām al-layl to refer to praying without going to sleep at all.
I use tahajjud to refer to giving up sleep or rest for the sake of worship regardless of whether one has already slept or not. According to the scholar al-Mawardi (d. 1058 CE) this is a valid usage of the word tahajjud and it is the first opinion he mentions about it (in his commentary on verse 17:79, at volume 3, p. 264 of the Dar al-Kutub version of his tafsir).
The two terms can be considered synonymous since there is no compelling evidence in the Quran or the Sunnah to differentiate between them. Since tahajjud means “keeping vigil”, any form of keeping vigil for worship can be considered to fall underneath it. Some scholars prefer to differentiate between qiyām al-layl and tahajjud and they are free to have that opinion. In my view it is not important what we call it, the point is staying up at night for worship.