I haven't practiced in Kentucky for more than 2 decades but I think your problem is the existence of two different systems and a confusing set of interplay.
When child protective services gets a report about a child who is being injured or is at risk of injury, they categorize the case as either one of abuse or neglect. If they take your child and go to court or go to court to ask the court to take your child, the type of case they file is called a dependency case. The child is alleged to be dependent on the Commonwealth for care and protection because the child is being or is at risk of being abused or neglected. The judge kept saying the dependency because that is the type of case that was before the court. It was a dependency case, not a criminal case or a civil suit for damages or an eviction.
The case worker said he found neglect and neglect is on your record because they brought a dependency case because of neglect (rather than abuse).
If you pled guilty and just went about working on getting your child back, you admitted to the neglect that brought your child into the dependency case.
It sounds as if the deadline would have already passed but you should contact a local dependency attorney to see if you can still file for a fair hearing to try to have the neglect finding reversed. You should also look into whether and under what circumstances being found to have merely neglected your own child can be used to prevent you from working in the health care field. The licensing board for your certification might be able to tell you.
I have linked below the U.S. Dept. of Health and Human Services' Administration for Children and Families' Child Welfare Information Gateway, which has links to federal policy and state laws and regulations about abuse, dependency and neglect.
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What most people, including attorneys, don't know is that there are two ways to end up on what is called the "central registry." The first is through proceedings in juvenile court like the one you descibed. The other is through an administrative process un a federal law called CAPTA. All juvenile proceedings in Kentucky are confidential and your employer should not have any access to those records or what is pending before that court. If they have been given that information, it has been done so illegally unless order by the judge.
What may have happened is that the DCBS made an administrative finding of neglect before anything was filed in court. Then when the case was disposed of in district court, there was a stipulation to dependency. If that is the case, the court finding takes precedent over the adminsitrative finding and your name should have been removed from the "central registry."
You should consult an attorney and he/she will need a) any letters you have received from the DCBS particulary any letter indicating that a claim against you was substantiated; and b) the adjudication order from district or family court.