Bring a copy of the page of your own car's insurance policy showing the amounts and kinds of coverages you already have.
Car rental company policies and rules vary, some require that non-citizens* buy one or more of the insurances and coverages.
You have to do some digging to find the rules and policies for each company but the easiest way is to look in travelocity.com for major car rental companies (you don[t have to book there). As a non-citizen you probably should not rent from little local companies. As a non-citizen you probably should not use Priceline or Hotwire, since it is easy to get into an argument about insurance. In my opinion the car rental company must refund your money if it wants to impose rules that do not show up in Priceline/Hotwire booking web pages but you really don't want to waste time arguing.
Collision is typically $20. per day, sometimes less, sometimes more.
Alamo has a choice of levels of collision coverage you can buy from them and you pay the rest of the damage. Rental companies also offer liability insurance for people/objects you hit, and separate insurance for injury to passengers in your car.
With most credit cards that offer car rental insurance and collision waiver, your own car's insurance, if you have enough of it, pays first. This might still result in higher insurance rates next year for you. In your case, if your own insurance doesn't cover for your being out of the country, the credit card insurance will cover it all. (Check the rules of your credit card.)
At least in the U.S. car rental agents try to push you into buying their insurance and coverages. Do not take the agent's word that you need insurance. Florida law does not require that you buy coverage from the rental company. If you want the coverage, buy it. If you don't want the coverage and the printed rules don't require it, keep saying NO.
Disney hints:
http://members.aol.com/ajaynejr/carrent.htm
The word you are looking for, in English in the U.S. is "deductible". Typically, after your own insurance pays first, all that is left for the credit card company to pick up is the deductible which is typically something like $200. or $500. in the U.S.
*If I were visiting Sweden, I may well be required to buy car rental insurance from an agent there, again it depends on the company.