Don't forget superposition, it really simplifies analysis sometimes.
I suppose the most fundamental way to look at vibrating strings near a pickup is to say that the permanent magnets that are part of the pickup setup a static magnetic field. Then, a ferrous but stationary string causes distortions in the field. A vibrating string causes vibrating distortions, for want of a better description. The coils don't care about the static part, they only see the changing part, the vibrating part, which in this case is all mixed up with a stationary part that doesn't matter. It's really difficult to visualize just the changing part.
But there's a much simpler way to look at it. The string does become magnetized (it is correct to say that.) The resulting field (what we called a distorted field above) is just the sum of the permanent magnet and the magnetized string. That's superposition. Easy Peasy. If the string is vibrating, then the resulting field is just that of the permanent magnet summed with that of the vibrating string. But we already said that the coil of wire doesn't care about the static part, just the changing part. From this perspective, it's easy (ok, less difficult) to visualize the changing part: it's that of the vibrating string. In other words, you get the right answer if you just consider the magnetized string and ignore the permanent magnet entirely - except for the fact that it magnetizes the string. After it's done that, you can ignore it.
The only leak in the bucket is a very tiny one. The degree to which the string is magnetized depends on where it is within the field of the permanent magnet. So, if in the course of vibrating it moves farther away from and then closer to the magnet, its own magnetization changes slightly. But it's what physicists and engineers call a secondary effect. It's small compared to the whole.
