My advice would be to cut up your debit card. You have much less protection under the law in the USA than a credit card and while your arguing about it your money is out there and not theirs.
Now an exception to this and maybe not a good one because I don't know the legal protections is an ATM card. I've had very good luck using these and the fees are lower than a credit card in my experience. I only use these at a bank or what I think is a reputable business.
I would not use either a credit or debit card except in a bank or a hotel that I am staying at. I have heard several horror stories of people using them in other locations. Having said that a few years ago I went up to Chetumal, MX and the only place I used my credit card was at a moderate size hotel. The bank notified me of suspicious activity which I confirmed was not mind. No harm no foul but had I used a debit card I'm not so sure it would have had good results without a lot of effort.
I would say that San Pedro is reasonably safe to use a card but you must always check your transactions very frequently to make sure all is well. Sometimes it is not the business but a less than honest employee who gets your information. Beware.
Also sometime these transactions show up months after you have used your card. Frequently check your transactions no matter where you are.
Somewhere on a beach in Belize