Is it permissible to count the amount paid as a loan to a poor person from a vow or zakat? Is this person considered eligible for them? And does not remembering the exact wording of the vow permit paying its value in cash?

It is not permissible to waive a debt with the intention of it being Zakat, nor to absolve a debt with the intention of it being a vow (Nathr). This is the fatwa we adhere to.

If your relative is poor and unemployed, then they are among those eligible to receive Zakat.

The ruling concerning a vow (Nathr) is similar to that of Zakat: it should not be given to someone whose maintenance is obligatory upon the vow-maker.

A vow must be given to the poor, unless there was an intention at the time of making the vow for the vow-maker and their family to partake of it.

If the vow-maker forgets whether the vowed item was to sacrifice a sheep or to give charity in money, then the safer course of action is to do both.