Is the supplication, "O Allah, I pledge to You to abandon this sin if You forgive me," considered a vow or an oath that necessitates expiation, given that the speaker intended it not as a vow or an oath, but rather as a supplication seeking forgiveness?

The supplication in the form mentioned in the question should not be made, because repentance is obligatory, and it is not appropriate to condition it on such a stipulation. The correct approach is to resolve to repent and ask Allah for help and forgiveness. This form is neither a vow nor an oath.