Bathroom reno - cutting floor joists!?

Answer one is correct, it's done all the time. The problem is with what they consider right. If it's an older home the joists may not be the full 12 inches required today and a toilet drain is at least three inches. That size hole can considerably weaken a joist, especially if it's under a support wall. I'd ask them to explain the building code for this situation. Or better yet call your local code enforcement office for an explanation. Your contractor did pull a permit, right?