"It was better than I expected," said her mother, Deadra.
She is now able to get out of bed and eat solid foods, reports CBS News Correspondent Bill Whitaker.
Lynch's father, who arrived Sunday from West Virginia along with her mother, two siblings and a cousin, said the family was happy to discover that her condition was not as bad as they had feared.
"It really felt good once we'd seen her and seen the spirits she's been in," Gregory Lynch told reporters during the family's first news conference at Landstuhl Regional Medical Center. "We knew she was going to be all right."
Deadra Lynch said her daughter was "real cheerful," emphasizing that the family was not pushing her to discuss her harrowing experience in Iraq.
"When she's ready to tell us something, she will," said her father.
Lynch, a 19-year-old Army supply clerk from Palestine, West Virginia, was captured March 23 after her 507th Maintenance Company convoy was ambushed in the southern Iraqi city of Nasariyah. She was rescued April 1 from a hospital in the city by U.S. commandos and airlifted to Germany.
Lynch does know that some of her colleagues lost their lives in the ambush that led to her capture and that others risked their lives to save her, reports Whitaker. It's unclear whether she had asked about their fate after being rescued or already knew about it.
"She doesn't really think that she's a hero, but she's a hero as well, too," said her brother, Gregory Lynch Jr.
The family says she survived because she is strong of body and of mind.
Doctors say Lynch is in stable condition in intensive care, where she was being treated for a head wound, an injury to her spine and fractures to her right arm, both legs, and her right foot and ankle. Gunshots may have caused open fractures on her upper right arm and lower left leg, according to the hospital.
Doctors hope Lynch will be well enough by the end of the week to endure the long flight home, Landstuhl spokeswoman Marie Shaw said. CBS News Correspondent Stephan Kaufman reports the family will stay in Germany until Lynch leaves.
Lynch underwent back surgery Thursday to correct a slipped vertebra that was putting pressure on her spinal cord. Since then, she has undergone several more surgeries to stabilize the fractures.
In a clear sign of progress, she got out of bed and sat in a chair for four hours Monday and was sitting again on Tuesday, Landstuhl hospital commander Col. David Rubenstein said.
He said she has begun physical and occupational therapy and has eaten solid food for the first time since her ordeal — turkey with gravy, French fries and carrots for dinner Monday.
"Pfc. Lynch is doing well and is in excellent spirits," Rubenstein said at the news conference.
Lynch's family appeared solemn during the 15-minute news conference outside a guest house on the hospital grounds, smiling only at times. They wore yellow ribbons to recall the fate of U.S. soldiers still in captivity or missing in Iraq.
Gregory Lynch Jr. said his sister was "the same person" he knew before.
"She's just a bit tired now, and trying to get better," he said. "She's very strong, strong in the head. She's very determined."
And she wants to go home.
"That's the only thing she's been asking," her brother said.