I do not know if it is true or not, although perhaps another attorney can help clarify - I have not personally seen a livescan report, although I have taken them more than once. Looking at the source where you obtained the information, it seemed that the person was advertising their personal background check service, so that throws a little doubt on the article's credibility. I definitely do not know of any California law that "mandates" only certain records be shown in a livescan report. It is true that the reporting services for Calfironia are not the most efficient, though.
A livescan can potentially include anything in the "State summary criminal history information" which includes the master record of information compiled by the Attorney General pertaining to the identification and criminal history of any person, such as name, date of birth, physical description, fingerprints, photographs, date of arrests, arresting agencies and booking numbers, charges, dispositions, and similar data about the person. (See California Penal Code section 11105).
If you wanted to be sure whether you saw a conviction or not on your livescan, you could request a personal Record Review, paying the associated fees. This would probably be similar or the same as what other potential parties would see in your livescan report.
Assume it reveals everything you know about and be prepared to address the issue. You should reveal the fact that you had a DUI. Accept responsibility for it. Explain what steps you have taken to correct the behavior that caused you to get the DUI and how youw know it will not happen again. How remote in time was the incident, etc. A misdemeanor will show up on the live scan. Think about it, we don't want people under the influence to look after our children. But, the government understands that we all make mistakes. They will give you the opportunity for employment if they see that you were up front about it, that you acknowledge the wrongdoing, have corrected the behavior, and have not picked up any new like or similar offenses been government.