But that's why it's so confusing. Did they purposefully leave off the reference to indicate Watts, or did they mean volts (as I'd guess audio engineers typically do when talking about amplifiers).

No, it isn't Watts if there's no reference. It's a ratio of powers. i.e. amplification or attenuation.

To have an actual amount of power, you need to reference it to some particular amount of power, like dBm for power-ratio-relative-to-milliwatt.