Is it considered a lie for an employee to pay from his own money to compensate for an error in a financial amount that the customer was supposed to pay, and then record something else of the same value in the ledger without informing the manager?

If the error in the financial value resulted from an intentional act or negligence, then the woman is accountable for it. If the money can be returned without lying, that is the desired course of action. However, if harm is feared, then what you did is permissible, with the firm resolve not to repeat it; because lying is forbidden except for what has been exempted, and in veiled speech (ma'arid) and dissimulation (tawriya) there is a way out from it. Guaranteeing the financial error is not obligatory if there was no intentional act or negligence.