"the fly ash emitted by a power plant—a by-product from burning coal for electricity—carries into the surrounding environment 100 times more radiation than a nuclear power plant producing the same amount of energy"
The quote isn't actually comparing coal waste to nuclear waste, it compares coal waste to overall nuclear power production. No nuclear power plant can be 100% shielded from all escaping radiation, nor can containment of radioactive waste, unless you build a vault so thick that no gamma ray can pass entirely through it, which is statistically practically impossible. Therefore, nuclear power production leaks a very small amount of radiation. At a certain distance from the reactor or waste storage, the radiation emitted from the reactor or waste drops to a point where it becomes indistinguishable from local background radiation. It is in this range that the radiation has added measurably to the environment. In a proper plant design, this range is within the plant's perimeter, and ideally within the containment structure itself.
So, no, it isn't saying escaped/released nuclear waste, it is referring to the radiation emissions from nuclear power production, which some percentage of will pass through the available shielding.
At some point, when I have some free time, I'll try and work out the relative radiations of actual waste products, out of curiosity. For reference, coal fly and bottom ash, together, release 5-6 (up to 8) picocuries of radiation per gram. A modern coal plant produces fly ash that leaves the stack at about 100g/MWh (going directly into the environment, uncontrolled) and produce 85kg/kWh of ash (recovered fly & bottom ash). Therefore a typical coal plant producing 3.5TWh/year creates waste emitting 1.8 curies/year, or 66.6 GBq (GigaBecquerels)/year. I don't have numbers handy for the waste products per power produced for nuclear.
Of course, I'll agree that this is somewhat overstating the case. When stored as a unit in a giant landfill, the vast majority of this radiation will never leave the landfill, because the material surrounding it will act as shielding. Only radiation emitted within, say, the outermost 10 meters or so will have any significant chance of reaching the outside world.
But the point isn't really that coal ash is a particularly dangerous radioactive substance, it is that we analyze nuclear power to an extreme but other sources of power get a free pass because they look easy to understand. When analyzed side by side, taking all factors into account, nuclear power, especially modern plants, should come out on top. At least until we have a significant fusion capability, or solar becomes significantly cheaper, more efficient, and lower in production toxicity.