You'd have to check with the authorities in Puerto Rico to see if there is a method of amending your birth certificate at this late date and whether you can have it amended remotely. I doubt that either is possible, but you should check.
In Florida, you can file a petition for name change and have your middle name changed to the way you currently use it.
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It may be possible, but without knowing how Puerto Rico operates when it comes to a question like this I cannot effectively advise you.
Mr. Rose is correct that in Florida you can petition to have your name changed, however, it seems from what you have mentioned that your name is correct on all other forms of identification, and that the PR birth certificate is the only issue.
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If it cannot be done in PR, consult an attorney about a petition for name change. I have been doing a lot of them because of the recent changes requiring a birth certificate for renewing a DL. The judges are granting them without hearing, but unless you are restoring a former name, a criminal background check, both state and national is necessary.