After Vader cut off his hand and told him that he was Luke's father, why did Luke just jump into the chasm? We know he eventually got away, but did Luke have a plan about how he would escape before he jumped, or did he jump because he lost hope and just happened to get saved?
Luke's plan was suicide.
From the new canon "Star Wars: The Empire Strikes Back: So you want to be a Jedi?" novelisation of the film
You want to be with your father. More than anything in the universe, this is what you want. It is what you’ve always wanted.
But Vader is not your father. Not anymore.
You let go of the platform and fall into the abyss.
And the Star Wars in 100 Scenes factbook
Despite his agony, Luke senses the truth of Vader's words. He has another choice - to die. He lets go and falls down the reactor shaft.
This is backed up by the film's junior novelisation
His voice is so hypnotic, Luke thought, and felt part of himself falling under Vader’s spell. But only part of him. Luke looked down into the deep shaft that seemed to stretch down to forever.
“Come with me,” Vader urged. “It is the only way.”
Luke looked directly at Vader and felt a certain calmness as he thought, No. It’s not the only way.
and from the film's official novelisation
With the calmness that Ben himself and Yoda, the Jedi Master, had taught him, Luke Skywalker made, perhaps, what might be his final decision of all. “Never,” Luke shouted as he stepped out into the empty abyss beneath him. For all its unperceived depth, Luke might have been falling to another galaxy.
and from the film's original script
Vader puts away his sword and holds his hand out to Luke.
A calm comes over Luke, and he makes a decision. In the next instant he steps off the gantry platform into space. The Dark Lord looks over the platform and sees Luke falling far below. The wind begins to blow at Vader's cape and the torrent finally forces him back, away from the edge. The wind soon fades and the wounded Jedi begins to drop fast, unable to grab onto anything to break his fall.
Having never read into this subject before watching the Luke's Scream scene just today, I always had the idea he had come up with a plan.
After freaking out about Vader's revelation and subsequent offer, anger grows in him. Realizing this is a dark path, he remembers Kenobi & Yoda's teachings.
He calms himself & uses the force to seek an escape.
He envisions the path to the ductwork and feels there's a way out, in similar a way as he had on his run of the Death Star's exhaust port.
That's why he didn't scream in the original. He had trust in the force. Force Trust Fall.