I have a valid US visitors visa. Does the "valid until date" stamp done at immigration include that date? Can I be in the US on that date or should I leave the US before that date?
US immigration does not have a "valid until" date stamp. They have an "admitted" stamp, which shows the date on which you entered the US. There is also a line with the words "class" and "until." Below that line, the officer may write or stamp a date.
Can I be in the US on that date or should I leave the US before that date?
The date shown below that line, if there is one, is supposed to show the last day on which you may leave the US. If you leave the US during the date shown, you are leaving on time.
Nothing in this stamp affects the validity of your visa unless it is a single-entry visa. It is therefore somewhat confusing to speak of a "valid until" date on the admission stamp. There is an expiration date on the visa, but that date has nothing to do with how long you are allowed to remain in the US (see, for example, Entering U.S. exactly on the day of visa "expiration date" on this site).
I wrote "supposed to show" because there is another way of checking how long you have been admitted to the US. You can look up your I-94 record at https://i94.cbp.dhs.gov. This record should show the date by which you're supposed to leave the US, and it should match the "until" date written or stamped in your passport, but there have been a few questions here suggesting that they sometimes disagree.