Is the money the questioner took from his workplace considered a debt upon him, and is he sinful for not informing the owner, knowing that the latter permitted him to take what he needs?

The store owner's statement about taking an advance on the salary when needed could be interpreted as permission for a loan, or as a dispensation to take without repayment. In this case, there is no blame on you as long as you do not exceed the conventionally understood need. It is more cautious to inform the store owner of the amount you took to ensure his forgiveness.