So the verses above are there, but the full passage of which you have only read part reads as follows (NET):
If men fight, and one strikes his neighbour with a stone or with his fist and he does not die, but must remain in bed, and then if he gets up and walks about outside on his staff, then the one who struck him is innocent, except he must pay for the injured person’s loss of time and see to it that he is fully healed.
If a man strikes his male servant or his female servant with a staff so that he or she dies as a result of the blow, he will surely be punished. However, if the injured servant survives one or two days, the owner will not be punished, for he has suffered the loss.
So the first rule shows the circumstances of a fight, with no indication of who started the fight, only what the outcome is.
The second rule also show the circumstances of a fight, also with no indication of who started the fight, only what the outcome is.
In both cases a person is injured to the point where they might die and if they die the killer dies also (definitely nothing sub-human about the rule for slaves here). And if the person injured gets up after a few days, the 'winner' of the fight has to pay in some way [right now I am getting a sense of deja-vu - I think I pointed this out to you a few days ago, so clearly you haven't been paying attention to what is being said]. There is no distinction here: In both cases someone has to pay.
The ONLY difference in application between the two is how the payment is made. In the first instance, the victor pays the victim. In the second instance - the net effect is the same. The victor pays the victim, but in this instance, the victim is in debt to the victor so money goes back to the victor immediately, but the victim has had time off work and that cannot be held against him in any way.
The parallelism ought to be obvious and also that the servant is not mere 'property' because if he were the master would be entitled to extend the debt for loss of work...
And on top of that a few verses later the servant can be given their freedom if they were maimed in any such fight.
By using your logic, You can beat your neighbour with impunity, as long as the neighbour lives.