Can I estimate a cell capacity (mAh) with the amperage and the time of work?
Yes, you should be able to get a reasonable estimate of the battery's capacity as it is now - not what it was brand new.

It runs at 19 volts at 0.7 amps for 1/2 hour.   So: 19 V x 0.7 amps x 0.5 hour = 6.65 Wh 

6.65 Wh at 19 volts after it has been boosted from its 12 volt nominal state.  You assume an 85% efficiency which is reasonable which means that you lost 15% in the boost, so 6.65/.85 = 7.82 Wh from the battery pack.

7.82 Wh for a 12 volt lithium 3s battery is really at 11.1 or 10.8 volts nominal.  Since we don't know which one, and it doesn't make a big difference either way, I'll use 11.1 volts as the nominal voltage so: 7.82 Wh ÷ 11.1 v = 702 mAh. 

Remember that it is a 3S2P so 702 ÷ 2 = 351 mAh per battery. 

This seems a little low to me, so perhaps I have made a mistake somewhere, but I don't see where.

A way to confirm is to use the powerbank to charge a battery that you know the capacity of.  If it charges that battery to 50% for example, you can then add the 15% loss for the boost to calculate the powerbank's capacity.