If I have a booster capable of reaching LEO, and want to launch it from a sailing platform in the middle of the South Atlantic (on the equator) do I have to notify some agency or state beforehand? Can it be mistaken for SLBM launch?
It seems the host country of the company requires a permit. This has been done by only one company that I'm aware of, Sea Launch. Sea Launch was licensed by the FCC, as it was managed by Boeing. If it's not a company, then it would be the host country. And quite frankly, if a person don't have citizenship of any country, then they seem unlikely to be in such a position. In any such case, the UN should be notified, along with any country over which a piece of the rocket might drop a stage.
It could fairly easily be differentiated from a ballistic missile, long before it would get to a point of any real danger. Most rockets burn out after around 10 minutes, while it would take a full 40 minutes or so to get across the world. Once the rocket is burned out, it's very easy to tell, and I suspect the profile is radically different even for a rocket not yet burnt out. You should also notify whatever entity manages the air traffic in the area.