The safest way for a pedestrian to cross a road is a location where there is the greatest opportunity to avoid a collision at any speed. That means minimizing number of directions you need to watch for traffic, and maximizing the likelihood of being in the line of sight of drivers. That means you want to cross away from intersections.
Crossing between intersections means that as a pedestrian you only have to be concerned about traffic from two (or even just one) directions, and for oncoming traffic you will definitionally be in the direction the drivers are facing.
Crossing at intersections means as a pedestrian you are having to watch for traffic from more directions, including directly behind you, and traffic approaching the intersection has drivers who are necessarily going to be having to look at places other than directly in front of them in the case of traffic coming towards you on the street you are crossing, and traffic coming from the other streets may not by physically able to see you on the intersecting cross street (from their PoV) prior to actually reaching the intersection.
Hence crossing between intersections is safer because it reduces the likelihood of any collision, as it's easier for everyone involved to be aware of everyone else.
Speed of a pedestrian vs vehicle collision is much less of a safety factor than just not having the collision at all, because the difference in speed between "walk away" and "going to hospital" is very small - well within normal intersection speeds. At higher speeds of course the likelihood of going to the morgue skyrockets, but when considering the safety of "low speed" collisions it's important to consider a "low speed" collision that is minor for an adult is still easily able to kill a child, and the speed _required_ to kill is not that high as demonstrated by multiple pedestrian vs cyclist collisions that have killed people (I think generally older people or just really bad luck but its just important to recognize that the "serious damage to soft and squishy people" is way lower than people think).
