Does what I did, making a covenant with myself to abandon sins, television, music, and major transgressions, while obligating myself to pay a sum of money to charities, count as a vow? Is it permissible to pay an expiation to break this vow due to the difficulty of fulfilling it? And what are the implications of forgetting this vow since 2004 regarding past actions and sins without having paid?

What you did is considered a "yamin al-lajaj" (a rash or contentious oath), and the person who makes it has the choice between fulfilling what they swore to or offering expiation for the oath (kaffarat yamin). Abstaining from what is unlawful (haram) is an obligation by the very essence of Islamic law and is further emphasized by an oath. Breaking such an oath leads to sin from two aspects: committing the unlawful act and breaking the oath. If you break the vow, you must repent and offer expiation for the oath, or donate the mentioned amount to charities, because a covenant with Allah is considered an oath. The expiation for an oath is feeding ten needy persons, or clothing them, or freeing a slave. If one cannot find any of these, then it is fasting for three days. If you did not intend to listen to or look at the unlawful things that are ubiquitous, and you averted your gaze or attention from them, then you do not break your oath merely by incidentally hearing or seeing them.