There is nothing wrong with Betadine or peroxide when used correctly. Betadine in particular needs to be diluted; the 9:1 would be good. Both should be rinsed from the wound with saline after use. They both are disinfectants. Both also kill new, healing tissue. In addition, Betadine has iodine, which can cause poisoning in overdose, and the iodine is absorbed by wounds. Thus, they should generally not be used after the first cleaning; saline is better if more cleaning is needed, unless there is necrotic tissue present. Dilute peroxide is a good approach for home cleaning of necrotic wounds (which of course really should be under a vet's care.)
Very much agree with dlhunicorn on the bath, by far the best way to get dirt and germs off, and to assess. I'd probably put a tsp or so of Betadine in the water rather than dish soap, if I had it, but either is fine.