How does companding compress dynamic range? Regarding when you 'compress' the signal, I typically see diagrams like this:

And the companding curve:

I can only see how the companding curve achieves a higher average SQNR by rounding down higher amplitudes more than lower amplitudes when making the quantization, according to a normal PDF, making the quantisation more efficient. I don't see how it reduces the dynamic range of the signal to the 'compressed dynamic range'. The only dynamic range compression I can see is that when the higher amplitudes are expanded again, they'll be expanded to the figure they were rounded down to during quantisation, reducing the upper dynamic range, but not the lower.
The dynamic range compression appears to be referring to the dynamic range of the signal after the compressor and before the expander, but this doesn't make sense. How can PCM samples have a dynamic range that isn't anything you want it to be.. and as the samples are uniform, surely a high rounded down amplitude now has a higher relative amplitude (sample 255/256 for instance whereas it was originally rounded down much more), and of course a lower amplitude is now a much higher amplitude.
What confused me is that there is a correspondence between 1.0 input amplitude and 1.0 output amplitude, which doesn't compress the dynamic range at all.
I hope that the key word in the output is 'relative' i.e. relative to the output range and it's actually mapping it onto a smaller dynamic range, though it looks like it's the same range. It would make sense to me if the dynamic range compression comes from the dynamic range that the samples are being mapped to is smaller than the input, and the companding curve is just a way to do this more efficiently than linear.
Why is the companding curve even called 'compression', surely the compression part is mapping it to a smaller dynamic range, or when using digital compression, the compression part is the reduction of bits in the PCM sample, and you can compress with linear companding, which wouldn't be considered compression at all if you were defining compression to be non uniform quantisation.
Companding is used when you want a signal to noise ratio (SNR) that's largely independent of the signal level.
Quantization noise is constant, so if the signal level varies, so does the SNR.
Let's consider an audio signal that's quantized at 8 bit. The quantization noise sits at -53dBFS. If you have a loud signal, i.e. -10dBFS you have a healthy 43 dB SNR. However, for a soft signal at -40dBFS, the SNR is very poor at 13 dB.
If you compand you can get an SNR of maybe 30dB for both cases, which is much better. The audibility of the quantization noise is NOT a function of the absolute noise level but the noise level relative to the signal. You can tolerate more noise, if the signal itself is louder since the louder signal masks more of the noise.