You're certainly on the right track. I'll review the situations as you stated, so that the comparison is clear.
1) In Minkowski spacetime we are interested in comparing two families of observers, the inertial ones and the uniformly accelerated ones. As you clearly stated what the inertial observers consider to be vacuum, i.e. absence of particles, the accelerated ones measure as a thermal bath, i.e. a thermal distribution of particles.
2) In Schwarzschild spacetime it is exactly the same. What inertial observers measure as vacuum the uniformly accelerated ones identify as thermal bath. The difference here is what constitutes inertial and accelerated motion. As per usual reasoning, in general relativity inertial movement is free-falling movement in the gravitational field. Therefore in Schwarzschild spacetimes the family of inertial observers are the ones falling towards the black hole. The uniformly accelerated observers here are the ones who manage to keep a constant distance from the event horizon, the acceleration being there to counter the gravitational pull.
What seems to be the source of confusion is that you are comparing one observer near the black hole with one far away. If both are in stationary orbit then both are accelerating and would see thermal bath, the only difference is that the one near the event horizon needs to accelerate more, and therefore sees a higher temperature than the one far away, just as in Minkowski spacetime. So what you are calling Minkowski observers in Schwarzschild, the inertial observers, should be the ones in free fall. Those really do not see radiation. You can check that the comparison holds because accelerated observers in Minkowski and stationary observers in Schwarzschild are confined to outside the horizon, while inertial observers in both cases have access to the whole spacetime.
3) As a side note, the previous case is not really the Hawking radiation, and in the literature it is usually called Unruh effect in Schwarzschild spacetime. What is called Hawking radiation is slightly different. What Hawking did was to consider an initial time where space is essentially Minkowski, but with some dust scattered. After enough time all the dust collapses into a black hole. What he found out was that one unique family of observers, who manage to stay outside the horizon, will eventually see thermal emission from the black hole.
In essence, the Unruh effect is about a single stationary spacetime and you are comparing two different family of observers. The Hawking radiation is about an evolving spacetime and what a single family of observers measure. The result are indeed connected, but different. In the stationary Schwarzschild spacetime we are in a equilibrium state, the stationary observers in orbit see radiation being emitted from the horizon, but also radiation going in. For Hawking's case the spacetime is evolving and the state is not in equilibrium. Those observers far away see the black hole emitting radiation but none going in (which should be the case, as there was no radiation outside to begin with).