Yes, it's a conviction. As far as I know a wet reckless--alcohol-related reckless driving-- is a misdemeanor, and less serious, with less serious consequences, than a DUI. But it is a criminal offense.
Not legal advice as I don't practice law in California. It's just my two cents in response to your question, based on general principles of law. If you need legal advice, please consult a lawyer who holds California licensure. That's not me.
As a general rule, being truthful up front is probably better than lying and hoping they don't find out about it later. So the answer to that question is, technically, yes. However, if they don't want an explanation, you can probably just get by with that. A "wet" is a type of reckless driving, but it is also considered a subset of DUI offenses. The choice is really up to you as far as what you want to call it.
Wet reckless is reckless driving involving consumption of alcohol. When you plead in court, the prosecutor likely asked you to admit on the record that there was a factual basis that alcohol was involved.
Legal disclaimer: This response does not create an attorney client relationship and is offered for informational purposes only. Only a lawyer fully versed on the facts and circumstances of your case can properly advise you on the case.
Here is the kicker, depending on when you got your conviction and whether you are eligible for it, you can expunge (or dismiss) this case. It would show up on your background check as a dismissal. Your arrest would show, your conviction would show and then your dismissal would show.
You can do this, but it takes some time...