Neither is true. 9kHz, with two sidebands, means that the transmitted audio is limited to 4.5kHz, which is way too low to sound good. It was this, and not the noise, that made it sound much worse.
So there were two reasons for the low audio fidelity of AM broadcasting, and noise was one of them, with the contention between multiple broadcasters for the narrow available bands being the other.
A non-obvious aspect of medium and long wave AM broadcast is depending on weather/atmospheric conditions a signal can propagate much further than its output power would suggest. This means a distant station on the same channel as a near station may end up in contention at certain times of day or random conditions. Solar flare? Suddenly stations a hundred miles away are overpowering local stations or just adding a lot of noise.
Medium and long wave is also susceptible to local EM sources like switching power supplies and electric motors. So you can get the double whammy of local noice and distant stations adding additional interference to local stations.
The only downside to this is that listeners on adjacent stations hear a slight "monkey chatter" from the overlapping sidebands.
This is one of many reasons why station frequencies are never allocated close to stations which are physically close.
You only need glance at the waterfall display on a good SDR receiver to see that the actual audio bandwidth is often much wider than the channel spacing implies.