This video compiles all the variations of the original intro, including every intro "gone wrong."

0:00 Peter Falls Down
0:49 Dancer Says She's Pregnant
1:48 Vinny Replaces Brian
2:22 Cleveland Replaces Mort
3:23 Stewie Forgets Lines
4:15 Peter's Credits Rap

Correction - At 0:48, S10E1 is actually "Lottery Fever," not "Lottery Ticket." Sorry about that.