Scott, you can "hold" whatever you want. The fact of the matter is that an infraction is a "public offense" and a "crime" under the Penal Code.
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And there's this:
The terms "crime" and "public offense" mean the same thing. They include felonies, misdemeanors, and infractions. (Hamilton (1986) 191 Cal.App.3d Supp. 13.)
And, then, from the California Attorney General:
Kinds of Crime (Pen. Code, § 16)
1. Felony (Pen. Code, § 17, subd. (a))
"A felony is a crime that is punishable with death, by imprisonment in the state prison, or notwithstanding any other provision of law, by imprisonment in a county jail under the provisions of subdivision (h) of Section 1170."
2. Misdemeanor (Pen. Code, § 17, subd. (a))
"Every other crime or public offense is a misdemeanor except those offenses that are classified as infractions."
3. Infraction (Pen. Code, § 19.6)
"An infraction is not punishable by imprisonment. A person charged with an infraction shall not be entitled to a trial by jury . . . [or] to have the public defender or other counsel appointed at public expense to represent him or her unless he or she is arrested and not released on his or her written promise to appear, his or her own recognizance, or a deposit of bail."
The only time a person would be taken into physical custody for an infraction, thus possibly having a public defender assigned, is if he refused to give his written promise to appear in court on a notice to appear (citation), and then only if he was subsequently refused bail or was not released on his own recognizance. However, failure to appear in court, even at a lawfully continued date, or to otherwise comply with the requirements of the citation received for an infraction, amounts to a misdemeanor or a Penal Code section 19.8 infraction. (Pen. Code, § 853.7.)
In other words, an accused who fails to appear may face a separate misdemeanor prosecution for that offense in addition to prosecution for the underlying infraction.
"Except as otherwise provided by law, all provisions of law relating to misdemeanors shall apply to infractions including, but not limited to, powers of peace officers . . . ." (Pen. Code, § 19.7.)