According to your Food and Drug Administration link, you do not need both the prescription and the note translated into English, one or the other should suffice (added emphasis mine):
In general, you should have with you a valid prescription OR doctor’s note—written in English—to bring medication to the U.S. The medication should be in its original container with the doctor’s instructions printed on the bottle. If you don’t have the original container, bring a copy of your prescription or a letter from your doctor explaining your condition and why you need this medication. Travel with no more than you need for your personal use during your stay. A rule of thumb: Bring no more than a 90-day supply of medication.
Go with your plan to bring a note, written in English, from your physician, and have it include the generic name of the medication. Without knowing what drug it is, antidepressants are generally Schedule IV, which are non-narcotic controlled substances in the US. These, according to the Drug Enforcement Administration, can be brought into the US when they are for your personal, medical use.
Unless a medication is prescribed by a physician and the quantify carried is reasonable (coinciding with the travel plans), US customs can confiscate substances which it cannot confirm are not narcotics.