This is no longer true in San Francisco (and possibly CA as a whole?). Pedestrians are allowed to enter the intersection when the don't-walk sign is already flashing, as long as they are out by the time it goes solid (in practice this means you usually will only do this if the flashing don't-walk sign is accompanied by a countdown timer). I believe this law change happened at the beginning of 2020.
As someone who is a consummate jaywalker, I love this, though I do think it's questionable from a safety perspective. Then again, is it? If someone is already moving in the intersection when the light turns green, oncoming traffic should be able to easily see the person. And they'll just be starting to accelerate from a stop, so (with the exception of idiots with high-perf engines who just need to show them off all the time) the cars will be moving pretty slowly. They can stop again just as quickly if they need to, and pedestrians still have a good amount of time to get out of the intersection.
This "already moving in the intersection" phenomenon is (I believe) what drives a semi-recent light timing change: the pedestrian signals at many intersections now transition to "walk" about 5 seconds before the drivers in that direction transition from red to green. This gives time for pedestrians to get into the intersection and be seen, before drivers start making left and right turns into the crosswalks.
When the "WALK" sign turns to a flashing "DONT WALK" (or their equivalent symbols) pedestrians are not supposed to enter the intersection.
Pedestrians already in the intersection can safely complete their crossing, but if you're on the sidewalk, you aren't supposed to step into the crosswalk.
It's basically the yellow light for pedestrians.
This is 100% completely ignored everywhere that I have been. Especially in big cities.