why ±10V is a common voltage range for commercial DAQ cards
It probably stems back from analogue op-amp circuits being powered from ±15 volt power sources. 10 volts peak would be a pretty normal undistorted or unclipped signal that could be produced from an op-amp on this power rail.
I think that using 10V instead of 1V on the output of the external
sensor board would give +20dB of SNR
Does a higher voltage in the transmission line provide better SNR?
Yes it does but it comes with a cost. Adding a gain-of-ten stage comes with an error and that is the resistors in the gain stage. They have a tolerance and they will add a gain error so, can you live with this error? Is the error a "worse thing" than the smaller SNR when not having a gain stage? Maybe it can be calibrated out?
I don't know if I should add more gain at the sensor amplifier to have
10V in the cables and then attenuate the signal on the ADC card, or
just output in the ±2.048V range of the adc (internal reference)
Adding an amplifier to boost gain comes with errors (as mentioned above) and then, reducing that signal at the ADC comes with the same sort of resistor tolerance errors so, you need to consider that against the backdrop of how far the signal has to travel (cable length). Personally speaking (having designed these sorts of systems), if the cable length is less than 50 metres and the bandwidth is basically audio then I'd stick with the lower amplitude signal. However, I know nothing about the EMC environment you need to operate in and this could be a significant factor in making a decision.
If it's a multi-channel system I'd also consider digitizing at the sending end and then make a high bit rate digital transmission to the far end of the cable. I've also got experience in this area too and so I know it works (96 channels sampled at 5 μs combined digitally and sent over 50 metres to a receiver).