Whose example finally compels Betty to rise and speak? What does she say? Why does Betty cry out with “great relief”?

It is actually Tituba's character that gets Betty to rise up and speak. Tituba is no fool. She knows that she is taking the brunt of the blame for conjuring the Devil. She is able to manipulate Reverend Parris into thinking he has the power to make her confess. She also deflects blame on to other people. This absolves her somewhat for her "crimes". Betty sees this strategy is worth pursuing so she follows the same pattern.

As she is speaking, Betty is rising from the bed, a fever in her eyes, and picks up the chant—the chant is echoed in the distant music of the dance in the forest—there is wind in the trees.
BETTY, staring too: I saw George Jacobs with the Devil! I saw Goody Howe with the Devil!
PARRIS: She speaks! (He rushes to embrace Betty.) She speaks!
HALE: Glory to God! It is broken, they are free!