Is what the questioner did—withdrawing from the first covenant and concluding a second one, then expanding his use of the internet beyond some clauses of the second covenant—considered a breach of covenant with Allah? And is it permissible for him to use applications and groups not explicitly mentioned in the second covenant if they are beneficial for his religion and studies?

Nothing is binding if the vow was not uttered. If it was uttered, then it is treated like an oath. One should look at the intention of the one who made the vow. If he intended not to use specific websites or programs and then broke his vow, expiation becomes obligatory for him. If he had no intention, then one should look at what prompted him to make the vow. If it was to avoid wasting time, he does not break his vow unless he uses something that wastes his time. If there is no intention and no reason for the oath, then it refers back to the implication of the wording. The expiation for an oath is feeding ten needy people, or clothing them, or freeing a slave, or fasting three consecutive days if one is unable to do the former. If the interest necessitates using what he swore to abandon, then he must expiate for his oath and do what is better.