To the OP, I can't tell you about Florida specific law but since you said you are from the Chicago area I would like to say I am Basset certified and have been working various alcohol related jobs in Illinois for the last 8 years, and you shouldn't have any problem at all using your state issued ID at any location as of 12:01 am on the day you turn 21. That being said, there are a few things to keep in mind
#1 Alcohol fines are HUGE. $500 for the first offense for the server along with $5000 for the bar/restaurant, and second offense they lose their liquor license. So they will make their own policies, and they are well within their right to enforce them. One of these include not accepting the "under 21 until" ID's even after the date. Although I can see a place not selling on the business day even if it is legally the next day, again that is a policy and not a legal distinction. The places I have worked have not made either of these exceptions though, we would let you drink anytime as of the day you turn 21
#2 The license you have expires 3 months from your 21st birthday. I will not and can be held legally liable if I were to sell you alcohol on an expired license/ID, so make sure you get it updated ASAP.
#3 If you have a passport I might suggest using it, as that big red Under 21 may be a turnoff even without it being technically correct. If you don't have a passport, get one and start exploring the world outside of the USA, it's pretty nifty
#4 If you get refused DO NOT GET UPSET. I know its a downer, but you do have a few options still left. You can always ask to see a manager, and if you are polite you will go a lot further. You can also always go somewhere else that may not be as picky, though in MK I understand BOG is the only option, it's just a short monorail ride to a lapu lapu... If you are unruly about it you will not get your way, and going back to #1, with such stiff penalties if someone doesn't think they should be serving you they absolutely will not, and if a guest gets mad I know I won't be serving them regardless of what may have happened.
#5 Have fun!
To those outside of the country, know this: In the states the drinking age is 21 and STRICTLY enforced (in some areas at least, especially tourist areas/cities). The fines for this are steep, and there are government groups that have no other task then to test alcohol sellers and see if they sell to someone with improper, altered, or underage ID. Because of this places have to educate their employees on different IDs and keep books that tell them of different identification security measures and ways to authenticate them. For this reason most places, especially those under stricter scrutiny, will only except three forms of ID. Those include ID's issues by state governments (drivers license and state ids), U.S. military ID and passports. Because so many countries have poorly made, easily doctored, hard to validate, or otherwise incompatible ID it is hard to educate on what is and is not valid, authentic identification from other areas of the world. I know where I work we do also accept IDs from the Canadian provinces/territories, but not from Mexico or anywhere else in the world.
Please note: I think US drinking laws are incredible stupid, over enforced and actually cause more harm than a socially progressive culture that doesn't demonize and thus under-educate/ill prepare its youth for intoxication. This does not change the fact that these laws are in place, the penalties stiff and the enforcement strict, so please don't try to argue with those serving you about how its 18/16/whatever where you come from. It is just one of those crappy things about traveling abroad, sometimes there are stupid laws you have to follow in order to more easily make the world a smaller place.