Over the last 18 years, I have carried rechargeable AA and weird sized (larger than AA) used for camera strobes and handheld lights, always packed carefully in my checked luggage. When the rules changed in the US to carry lithium batteries in carry-on luggage, I began carrying them on. Never had a problem at Cozumel when leaving until last year, when they wanted to confiscate them. After useless arguing, they finally suggested I take them back through security, and see if I could put them in my checked luggage. The agent said I had to put them in a bag (so I had to go buy a small bag at the little convenience store, and they had a baggage handler locate my luggage. I thought this was super dangerous, but I wasn't willing to throw away the very expensive larger batteries, so I went along with it. They actually dragged one bag out, I added them, and went back through security. When I got through Customs in the US, I pulled them back out of checked luggage and put them in my carry-on. WHAT A PAIN! Once in 18 years.
This year, I asked at the gate, and apparently Cozumel is up to date - at least this time. The person asking us about batteries BEFORE we got to the desk to check showed us the sign saying no batteries, and told us we couldn't carry them on. When I pushed back and explained the rule with US carriers, she asked someone at the desk and they said that indeed we needed to carry them on. Basically the left hand doesn't know what the right hand is doing - flip a coin. It's not going to stop me from going to CZM, but it is irritating.