By both of your definitions this appears to be a clock. It starts at twelve, not at zero, and it wraps when it gets back to twelve again, making it pretty useless as a measure of elapsed time.
For what it's worth I would define both a timer and a clock as something that ticks(/updates) at a fixed regular interval.
I get what you're saying, but have you never experienced a clock set to the wrong time? Is such a thing really no longer deserving of the title? What about clocks in different locales, are they not clocks for you but clocks for others? Is this demo a clock twice a day? You're the one claiming to be a pedant here.
For what it's worth I would define both a timer and a clock as something that ticks(/updates) at a fixed regular interval.