Owned the register mistake before anyone found it? That’s the kind of person managers remember.

Reporting your own error changes what the error is. Undiscovered, it is a problem with a hidden cause; reported, it becomes a fixable thing and a data point about you. Managers are not really tracking who never makes mistakes, because that person does not exist. They are tracking who they will hear it from first, which decides who can be left to run a shift alone.

When something goes wrong on your next shift, say it before the count does. State what happened, what you already did about it, then move on. Speed matters more here than apology.