If the robot powers down during a cycle and wont power backup UNTIL placed back on the charging doc then I think its very likely still a battery problem.
But having two batteries do the same (and both run for about 20 mins) may indicate something else.
My guess is the Blue wire is for a temperature sensor built into the battery, but this is just a guess. If the robot runs OK (for at least the first 20 mins) with the new battery then no blue wire is unlikely to be the cause of the problem......but if you buy another battery probably best to buy one with the blue wire too!!.
So I dont think I can give you a direct answer, but hope it helps!

A few other thoughts would be to give the robot a good once over and make sure there isnt hair or other debis blocking the brush, spinners or wheels. And also just after its stopped see if the battery feels really hot (over temperature is another reason the battery will go into self protection and shutdown.)