Why stress about the IAA giving you a JAA PPL? The fact that you get an ICAO PPL allows you to fly day VFR within the Contracting countries FIR (ie CAA will allow you to fly a G-reg aircraft within UK FIR but not across to Ireland!) In practise this is nothing more than local CFI checking your PPL and doing a check ride then off you go in a club aircraft.
If you are doing an Approved JAA PPL course in Florida, by definition it is a CAA/JAA course since they are the only ones approving out of Europe tests. Ergo you will come back with a CAA/JAA PPL and all the priviliges / limitations that entails. It probably behooves you to understand exactly what type of PPL you will get (FAA, JAA, CAA) and make sure that the training organisation is approved by the appropriate authority. For instance they may follow the JAA syllabus but may not be able to test for it and hence give you an FAA ticket!
If you are planning on doing the F/ATPL course route then it doesnt matter, entry requirement is an ICAO PPL, (JAA, CAA, FAA)
Hope this helps.