Betadine is better then peroxide. In short peroxide may help to kill the "bad" stuff, but it also kills the good tissue on the edges of the wound (which is why you'll see white-looking tissue after using hydrogen peroxide…..that tissue is dead ). A simple saline solution is your best bet to clean out any debris. You can irrigate the wound with saline and then use a triple anotibiotic if you wish.
My vet always told me neosporin was fine for small cuts and scrapes for dogs.
For disinfectant and irrigation of a deeper wound, I have always used witch hazel - doesn't sting. When I had went to the ER because of bad bite from breaking up a fight, I had a really bad puncture on my leg and told the ER doc I was using compression (Coban), gauze pads, witch hazel and neosporin and they told me that basically they were going to only change one thing about my regimen and that was give me wider Coban (only available at hospitals). They also said at the same time I should also leave the wound uncovered a 2 hours each morning and evening (including leaving off the neosporin).
Sideline question – is betadine or neosporin good for general anti-infections on small cuts and scrapes? What about bacitracin?
When mine get cuts I use betadine to clean it out, rinse that off then dab it dry, and put neosporin on it very lightly. Bacitracin may also work, but I'd ask the vet if there's any question.