My definition of violence is that it's the intentional causing of harm. If we look at, say, Merriam-Webster they define it as 'a: the use of physical force so as to injure, abuse, damage, or destroy', which seems to agree with me on the requirement of intent with the phrasing 'so as to'. But it also contains another kind of phrasing that is really insidious. If for instance, it had said 'and destroy' rather than 'or destroy', would you have raised an eyebrow? Because the difference between 'and' and 'or' here is huge, the former requiring that every single one of the listed conditions be met, and the latter requiring only any one of them.
So if I say that it is violence, then someone else says "Aha, so you admit to combat sports being abusive! What cruelty!". No, I did not. I only need, say, the 'injury' part to classify it as violence. However, the hypothetical opponent of combat sports in this instance will intentionally misread the 'or' as 'and', claiming that since I admitted to it being violent I have admitted to every single one of the conditions. They will, of course, not apply the converse, and insist that since I reject the accusation of abusiveness, then I can't say that it is violence, even though that is the implication of their interpretation. That would greatly inconvenience their argument.
I very strongly believe that combat sports are not inherently abusive, due to the factor of consent. Damage and injury are obviously permitted, but only as far as the referee is willing to allow it and the participants endure it. Those are what make combat sports different from assault (which is the most common comparison I encounter in these kinds of discussions: "Well, if we allow MMA, then we might as well allow people to bludgeon each other half to death in the streets!"), and ultimately what makes them civilized.
Surgery is an interesting thing to bring up. An incision with a scalpel is clearly harm, but the overall effect of a successful surgery is healing. Any landed punch is harm, but the overall goal is to win. The difference is that a surgeon will sew up the incision, reversing the injury caused, whereas the winning fighter won't do that to his opponent. Well, is not allowed to, likely can't, but probably also won't. Similarities and differences.
So that brings us to the starting definition. When fighters apply their art, is the intention to cause injury or to gain victory? Both, I'd say, but on different levels.