If one says, " It is my duty to do such and such" instead of saying "It is my duty towards Allah to do such and such", would his saying be considered a vow?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The statement you mentioned, "It is my duty to do so and so," is not an act of Nadhr (i.e., vow), and thus does not indicate obligation. "Not having liability and the non-effectiveness of the vow" is the original ruling on such cases.

Allah knows the best.