No, damaging a UFO does not change whether or not the alien mission continues or how many aliens show up at the terror site. In terms of score, shooting it down gives you a few points, but not as many if you do well in the battlescape against a landed one. However, unless you've been paying attention to the UFO waves, you don't know if the terror ship is just going to scout the area, landing once or twice, or is going to generate the terror site. Since the terror site has the biggest chance for giving you a negative score, it's best to prevent that from happening by shooting down the terror ships and assaulting them.
So Landed > Crashed > Terror Site, but shooting down ensures you don't have to do the terror site.