Is renewing the vow to abandon the sin after breaking it, along with the intention that the first expiation permits internet use and fasting for ten days, considered a valid intention? And is the first expiation, coupled with repentance, sufficient without requiring anything else, considering that the vow is for deterring oneself and not for seeking closeness to Allah?

The aforementioned covenant is subject to the ruling of an oath made in a state of obstinacy and anger. Therefore, you have the choice between fulfilling what you covenanted to do or offering the expiation for breaking an oath (kaffarat al-yameen). The covenant is dissolved by breaking it, and the expiation is not repeated unless you intended repetition or the wording of the covenant necessitated it. Inner thoughts about renewing the covenant have no consequence. You are required to offer one expiation for breaking an oath for breaching the first covenant, along with the obligation to repent from the sin.