To answer your question; you are correct in that children traveling with their parents only need an original birth certificate (not a copy) on closed loop cruises from the U.S. to the Caribbean, Bahamas, Mexico, etc. Parents need either a valid passport or an original birth certificate along with a government-issued photo ID.
This is the way it is right now - nothing has changed. We just did this two weeks ago with our daughter and her son, neither had passports, but did have their birth certificates.
Concerning the passports when going to Mexico, it's referring to flying or driving into Mexico, not cruising.
However, as others have pointed out, I also suggest to my clients that it's always better to have a passport. If you should ever have to leave the ship in a foreign country due to an emergency and have to fly back into the U.S., you have to have a passport. If you don't, then you'll have to go to the local U.S. consulate and get a temporary passport, which is alot of time and hassle.
But if you don't want to go to the hassle and expense of getting a passport, then none is required in your situation.