Not taking your Betta out is the best option, but for some of us with smaller tanks (like my 2.5 gal,) you have to take the fish out b/c it is much less stressful. That being said, LMac54, my fish, Seuss, used to be just fine with netting (before I knew it could rip his fins,) but within the past few months it was really stressing him out. Now I can distract him at the top of his tank and sneak the cup behind him, and whoosh! he's in before he knows what hit him! : )
Depending on what size tank/ bowl you have, and how much water you take out at a time, depends on how much old water you want back in the clean tank. For ex.; my 2.5 gal I do a 50% water change once a week, so there is still some old water left in it, (and I also have a filter,) so the little old water in the cup won't make much of a difference. But if you have a 1 gal or smaller (esp. unfiltered,) and are doing 100% water changes, that small amount of ammonia in the old water in the cup (to me,) would be just adding ammonia back to a clean tank, though it's not the end of the world, just try to get as little old water in the clean stuff as possible.
-Cupping is better (I'd say anything less than a 5 gal you should probably take the betta out, b/c it will be less stressful because the space you're cleaning is so small.) Though if you use a net carefully, you can net.
-If you do a 100% water change, try to get as little old water as possible in the new water, but if you leave some old water in the tank/ bowl, you can put the old stuff back in.
Sorry, I tend to be waaay over-detailed. : ) These are just my opinions, and what i've found work for me, but I hope this helps you a little.