If you really would rather be safe than sorry, you should follow the advice on the page you linked to, as it is advice I've followed in the past:
A valid prescription or doctors note is required on all medication entering the U.S.
Now practically we all know that's not really the case. Bringing in paracetamol/tylenol, or Imodium, or asthma medicines, etc, are generally waved right through. However many countries, including the U.S. as documented, technically require that documentation for all medicines so they have recourse for suspicious circumstances.
In the case of any medicines you're not sure about, it's definitely worth obtaining a doctor's note - it doesn't have to be a prescription, as a letter from your doctor should suffice. If you really must bring in your codeine-based medicines, it seems like a simple and sensible step to take.
The CBP give similar and more extensive advice on another page, which would presumably cover codeine under "potentially addictive drugs/narcotics":
If you need medicines that contain potentially addictive drugs or
narcotics (e.g., some cough medicines, tranquilizers, sleeping pills,
antidepressants or stimulants), do the following:
Declare all drugs,
medicinals, and similar products to the appropriate CBP official;
Carry such substances in their original containers;
- Carry only the
quantity of such substances that a person with that condition (e.g.,
chronic pain) would normally carry for his/her personal use; and
- Carry a prescription or written statement from your physician that
the substances are being used under a doctor's supervision and that
they are necessary for your physical well being while traveling.