It sounds like you recieved deferred adjudication. If this is the case, you will need to ask the Court for an Order of Non-Disclosure, which is granted in some cases when a deferred adjudication is completed successfully. The procedure itself is fairly routine, but the qualifications are complex and cannot be listed here. Talk to a criminal attorney - they should be able to tell you if you are eligible, even over the telephone.
A regular probation does not come off your record in Texas. If the judge told you that they were wrong. A deferred adjudication may come off your record via a motion for non disclosure. A lot of people believe that your criminal record is wiped clean after seven years like your credit history, but that is not the case. A while ago a prosecutor was not allowed to go back further then ten years to enhance a DWI. The law recently changed allowing prosecutors to go back further then ten years to enhance cases to the felony level.
Public data is a private company. You should go to the county clerk where you received your probation obtain a copy and go to a lawyer to get clarification.
A DWI stays on your record for the remainder of your life. As Herman said, the law changed and now the State is not limited to 10 years to use an old DWI for enhancement purposes.
You could not have gotten a deferred adjudication for DWI for an offense committed in 1990. They quit having deferreds available for DWI in either 1982 or 1984, thefore you received a straight probation with a conviction. (This avoided having your driving license suspended because you took a special class.)
Some people mistake the number of years a DWI is quote on one's record because of the limited number of years traffic violations are reported on your driving record.
That your friend's DWI did not show up is just luck. It means that either it was overlooked when the data was loaded; or, it was not reported by the county.
The following is not even an effort at legal advice - this is just common sense. You are right that the conviction is quite old. You are likely not the same person you were (not that you were a bad person.) However, if you are deceptive on your job applications and discovered, logically that could be worse because it would mean you are dishonest.