If the actual law doesn't say it's illegal, it's not illegal. Period.
If you're looking for a precise definition of what's legal and what's not legal -- don't get your information from the driver's handbook, your defensive driving instructor or a cop. Get it from the actual law -- anything else could easily be wrong or misleading.
Where did you find the words "due caution"? If they weren't in the law ... they're not law. (They could show up in a court case that sets a precedent and that makes them "law", but that generally only happens when the law was ambiguous somehow and needed clarification.)The California vehicle code does not explicitly mention this. (http://www.dmv.ca.gov/pubs/vctop/d11/vc21452.htm), However, it is not unreasonable at all to interpret due caution as not entering the intersection when you can safely stop.
Your citation of the actual law did not say this. Do you have a citation for this statement? I don't know the laws of California so I don't have any specific reason to doubt your statement, except that the law you did quote didn't say this, and I don't see it in the rest of that section.And certainly if you're still in the intersection when the light turns red, you're in violation, in a car or on a bike.