In a previous life, I too worked @ a financial institution. Although banks are federally & individually state regulated, they have different policies as to lost/stolen credit cards & debit cards; which they provide you when you receive your card.
If your debit card has a VISA logo: your liability can be affected depending on how long you wait to report the missing card; along with any $/day withdrawal limits associated with your card. Typically, you are on the hook for $50 for each card you lose, unless you have loss/travel guard for your cards.
The absolutely safest way to use/carry a debit card is:
Do not sign the card...where the signature block is located...in ink, print "See Photo Identification"
You must also carry a photo id with you; but this prevents anyone from using your card (assuming the clerks check the id).
Of course, you should check with your own bank. But, it has been widely reported that most theives will not bother attempting to use cards signed in this matter.