The arrests resulted in no criminal filings. That alone suggests the officer was mistaken. However, if you are arrested again on the same charge and the DA or City Prosecutor does decide to move forward with charges, the prior arrests will cast a long shadow over how lenient the DA can be in negotiating a disposition. I would anticipate having a requirement to attend AA if you are again arrested and finally charged on such conduct. The arrests do appear in certain background checks but do not show on others. There is no blanket answer.
If you are concerned about your past criminal record, contact a local attorney who can determine whether you are eligible for a sealing or an expungement. Sealing of the record occurs when the court physically seals your court file, which may not be opened without a court order. An expungement is the physical destruction of your file. Contact an attorney if you are concerned a future employer may run a background check.
The answer to your question is that, since drunk in public is typically a misdemeanor, it's not as bad as a felony, and better than having nothing on your record. Because the convictions with the court are a public record, they show up as both - a public record through the courthouse, and a criminal background check.
I hope that this helps, but please feel free to ask if you have any further questions. It's my pleasure to help in any way that I can. All the best to you.