The anti-satellite weapons are ground-based and are part of the playing close to the read line together with the spy satellites. The space weapons are when you have a nuclear head deployed in space with a few seconds to minutes ability to reach anywhere on Earth.
You seem to have some misunderstandings about orbital dynamics and nuclear weapons. Placing nuclear weapons in orbit wouldn't allow for striking anywhere on Earth within a "few seconds to minutes" unless you had an absolutely enormous constellation that would be unaffordable even for superpowers. You are forgetting to account for orbital parameter changes and then re-entry. In most realistic circumstances, a ground or sea based ICBM would be able to strike most points on Earth faster than a space weapon. Do the math on this, or you can try to simulate it in something like KSP.
Nuclear warheads also can't be stored for more than a few years and still be expected to work reliably. They require periodic intensive inspection and maintenance. Even though space launch costs are coming down there is still no feasible way to conduct such maintenance in orbit.
Check some of the old cold war designs, they definitely thought in that direction. Space weapons were prohibited partly for the reasons you mentioned and partly due to the political liability of militarizing the space.
