Betadine is povidone iodine 10%. Xenodyne is a veterinary preparation that is polyhydroxine iodine 0.5%. So related, but not the same. Xenodyne has a bit more oomph, but is difficult to get now. Povidone iodine is just fine, and is mostly what gets used these days by vets and ERs. Strength is important, though - you have to make sure you are using the right concentration, and enough of it.
Hydrogen peroxide isnt so good. How did you flush? We basically use a long, curved-tip syringe that we jam in the wound (ow), and flush with strong pressure 3 times. The first time isnt bad. The second time hurts a lot, and you want to punch something. The 3rd, some people cant do on themselves, because hell, that hurts ridiculously. (Which is why, in the ER, they have lidocaine in the mix.)
For your reading pleasure, heres an article on the requirements for effective flushing of cat bites with iodine.
https://www.medscape.com/answers/768875-60785/how-should-animal-bite-wounds-be-irrigated
With the quantity we use in flushing, you basically leak the solution for the next 12 hours or so from the puncture site. Thats a lot more than most people would do. But the up side is you deal with that inconvenience and a bit of burning/bruising for a few days, but no infections, and it heals up pretty quick. None of us ever use topical antibiotics for a puncture - we WANT that drainage! The antibiotic cream doesnt do anything for the deeper wound, either. Same thing when treating an abscess on a cat from a cat fight. I always keep the area draining.