Thank you for your question.
No, an arrest and a conviction are not the same thing. A conviction means his case was taken through conclusion, and he either plead guilty or was found guilty of the offense and was sentenced. A person can be arrested and then no charges pressed, charges dropped, or be found innocent, and then they would not be "convicted" of a crime.
Public intoxication for people 21 and over in Texas is a class c misdemeanor and punishable by a fine up to
$500. The fact that your son paid the fine makes me think that he was in fact, convicted of the offense, meaning it could show up on a background check, and therefore, he would want to answer "Yes" to such a question.
However, since I cannot be absolutely positive this is what he was convicted of, or if he was convicted at all, he needs to verify this. He can do so by getting the records from the Clerk of Court
in the county where he was originally arrested/charged, as criminal
court records are public. They should be able to tell hiim what the disposition of his case was.