From one observer's perspective, on all moving objects time will run slower. But from the perspective of one of those objects, it's the observer who's slow. And the answer seems to be that both are correct!
Then there are (correct) statements like the ISS runs slower than the Earth. Based on above how can this be true? Why not the other way around?
That's a good question, and it highlights the fact that special relativity describes far more than just motion at a constant speed in a straight line, and therefore the Lorentz transformations. Confusion about time dilation is almost invariably due to taking an overly simplistic view of special relativity.
The only way two observers can directly compare their clocks is if they are at the sample place in space. Two observers in linear motion can directly compare their clocks only once because by definition they can never meet again. That's why both can conclude the other's clock is running slow without fear of contradiction.
However accelerated observers can compare their clocks more than once - the obvious example is the infamous twin paradox (which isn't a paradox, but that's a rant for another day). When two observers can directly compare their clocks twice, or more, there can be no ambiguity about which clock has been running slow.
The ISS moves in a circle around the Earth. In principle I can put an observer hovering in space in the path of the ISS, so the ISS passes the observer once every orbit. That observer and the astronauts on the ISS can compare their clocks whenever they meet, and when they do all parties will agree that the clocks on the ISS have run slower than the clock held by the hovering observer. That's why we say the clocks on the ISS are running slow.
