There can't be a negative zero. zero is neither negative nor positive.
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The scale doesn't start at zero, so you need to compromise or you get a systematic error.
A positive number is greater than zero; a negative number is less than zero.
Positive zero error means, instead showing zero it shows some value more than zero. Hence positive. Suppose if it shows some reading say 0.03 units. then while correcting we have to subtract the above from the observed reading. So correction is adding negative error.
Non-negative means either zero or positive.
A positive value is a value that is greater than zero. A negative value is a value that is less than zero.