By choosing to enter your plea with the clerk rather than in court, before a judge/commisioner, you did in fact waive your right to a speedy trial.
California Vehicle Code Section 40519. (a) Any person who has received a written notice to appear for an infraction may, prior to the time at which the person is required to appear, make a deposit and declare the intention to plead not guilty to the clerk of the court named in the notice to appear. The deposit shall be in the amount of bail established pursuant to Section 1269b of the Penal Code, together with any assessment required by Section 42006 of this code or Section 1464 of the Penal Code, for the offense charged, and shall be used for the purpose of guaranteeing the appearance of the defendant at the time and place scheduled by the clerk for arraignment and for trial, and to apply toward the payment of any fine or assessment prescribed by the court in the event of conviction. The case shall thereupon be set for arraignment and trial on the same date, unless the defendant requests separate arraignment. A deposit of bail under this section does not constitute entry of a plea or a court appearance. A plea of not guilty under this section must be made in court at the arraignment.