As to your differences in approach, I'm inclined to side with your wife, but it seems none of you have found something that really works.
To a child, the fundamental award from a parent is attention. Thus, we reinforce the behavior that we grant attention. This holds for negative attention as well. So a general rule of thumb that is commonly cited is reward the desirable behavior, and ignore the undesirable behavior. This explains your experience of adding fuel to the fire. (A key difference to understand: as opposed to your title, your child does respond to your disciplinary actions, only not in the way you were hoping, and this is normal.)
But there are certainly beahviors, such as hitting, that we wouldn't want to pass unnoticed entirely. Another common take on that same idea is that for every one time you scold your child for doing something you don't like, you should strive to find five opportunities to praise your child for doing something you do like. This allows you to communicate clearly when you don't find a certain behavior acceptable, while at the same time making sure that the desirable behavior is more rewarding to the child (i.e. gets the child more attention from you).
Try to always use a personal language. Don't say "we don't spit on the floor in this house", because your child may not understand and value that rule. Instead, say, "when you act this way, it makes me sad" and "when you spit on the floor, I have to clean that up, because a clean house is important to me. And all the time I spend cleaning up after you, is time we could have spent playing." Those are adverse events that your child can relate to, and adhere to your rules not because they're the rules you've laid down, but because they make sense and seem desirable.
I generally find punishment to be both exceptionally ineffective (due to the disproportionate attention given to an undesirable behavior) and often cruel, so that's something I'll always recommend you avoid. Natural consequences that follow logically from the child's actions are more tolerable, to me. Don't say "you hit a child so I'm removing your baseball cards". Say "You can't hit other children. That makes them sad, and they won't want to play with you. And I can't have you play with other children if you're hitting them. Then we'll have to stay at home. I think that would be really sad."
Do let your child know what is acceptable. Avoid doing it in a manner that rewards the child with unproportional attention for unacceptable behavior.
When I say reward the behavior you want, I'm still talking about attention being the reward. I think bribes are also undesirable, although less so than punishment. I believe it teaches the child to expect a gift for something you want them to comply with out of free will, and I believe it's subject to inflation, where a parent might think have an idea of phasing out the reward over time, the child will likely come to expect the reward and instead require more (just as you might expect a salary increase once a year or so, for continuing to do the same job, rather than tolerating having your salary slowly phased out). The reward should preferrably be that you're telling the child how it makes you happy when he gets along well with others, and how you now have time to play with him because you don't have any cleaning up to do.
There is another rule of thumb that simply asserts that kids do well if they can. Examine the situations where your child is displaying an undesired behavior, and ask yourself if your demands are exceeding the child's abilities. That could be demanding an age-inadequate level of abstract thinking, to understand why an unspat floor is preferable, or expecting them to follow a verbal instruction of many words and complex structure that is simply too hard for them to parse. Or simply, expecting them to have the stamina to behave well after a day of many activities, or asking them to endure something that is simply too boring for them to handle.
As the adult, it is your responsibility that your interactions with the child are constructive. Do you need to talk in easier sentences? Do you need to provide more rest before a taxing activity? Do you need to guide your child through dealing with their feelings when they feel wronged?