In Texas you do not have to register as a sexual offender for the crime of public lewdness. In fact, it is a misdemeanor. You should ask this question in the Florida criminal defense forum because there may be something that triggers the registration there because of the definition of the crime in Texas.
This is the definition for public lewdness in the Texas Penal Code 21.07 is as follows: (a) A person commits an offense if he knowingly engages in any of the following acts in a public place or, if not in a public place, he is reckless about whether another is present who will be offended or alarmed by his:
(1) act of sexual intercourse;
(2) act of deviate sexual intercourse;
(3) act of sexual contact; or
(4) act involving contact between the person's mouth or genitals and the anus or genitals of an animal or fowl.
Good luck to you and your son.
The only way to the case off of your son's record is to have a writ of habeas corpus granted in the case. It is best to hire a lawyer to do this as it is rather complicated and you will find that the entities involved usually work more cooperatively with a lawyer (because the lawyer knows how to make them cooperate if s/he is competent.) Most likely a lawyer could get some of the records relating to the case to determine if your son has an issue to raise. The issue(s) must be of constitutional magnitude, such as ineffective assistance of counsel. In order to have gotten to the point of receiving "something" (community supervision, deferred adjudication, or jail time), you son would have been required to have pled no contest or guilty. The judge's involvement in the original case would have been most likely simply following the plea bargain that your son agreed to with the District Attorney's office. (It is slightly possible your son pled not guilty to the judge and the judge found him guilty of public lewdness.)
Herman is right that in Texas one does not have to register as a sex offender for public lewdness IF he only has one conviction. But every state interprets who needs to register by their own laws. Perhaps because your son's case was FILED in a district (felony) court, it appears to Florida that he has a felony. (This has been a problem in the past when cases have been reduced to misdemeanors rather than refiled in misdemeanor court.) If your main concern is registration (not getting it off his record - by writ), then I would consult a Florida lawyer and determine what the registration rules are. If I am correct about the misinterpretation of Texas laws, then you could get a copy of the judgment and sentence from Texas, and perhaps a letter of explanation from a Texas lawyer, and then provide that to your Florida lawyer to try to avoid registration.
You are right about "sex offender" stuff affecting all aspects of his life but the fact that he received punishment for lewd conduct could also cause him the same problems. When you think about both those sets of terms, they sound scary. Sadly, in Texas, the sex offender registration purpose of protecting the community has been so watered down by abuse of naming some "sex offenders" that it is of little value. (For example, if a couple has sexual relations and he or she is more than 3 years old than the other, THAT is a sex offense upon which one could be required to register as a sex offender in Texas. Ridiculous.)
I would seek a consultation knowing what I have shared with someone in Florida first to determine why your son is being required to register in that state when he would not in Texas.