When you ask some controllers to give power to the motor, they send a sequence of pulses and look for feedback. If they don't get the right feedback, they try a different sequence. Different controllers do it in different ways. That test is pretty well meaningless.Hi, the sequence of the lights displayed during testing were not in accordance with the testers enclosed instructions (which were somewhat difficult to understand as written in 'Chinglish'
If your battery is just worn out, I'd expect it to drive the motor with the wheel off the ground, and when you ride the bike, it would cut when you demand a certain amount of power. The same would happen with an under-rated battery.
Your description of your symptoms is unclear. The first bit makes sense: Your first battery lasted a year, which sounds about right for a crappy over-stressed battery. Then, you said you bought that 99999million AH one that immediately didn't work, so you got the tester. That makes sense too, since you bought a battery that can only deliver 5A continuous for your probable 25A controller. After that, you said that the 5A battery had worked for a year. That doesn't add up with the previous statements, and doesn't sound possible considering its rating.
AFAICS, all the Greenlance batteries would be suitable. The more Ah you get, the further you can go and the more power you can use while doing it. As i said before, the minimum requirement is that the battery should be able to deliver 25A continuous. 30A would be better, as the battery will be less stressed, so will last longer. You check the specification to see what is the maximum continuous current that the battery can deliver. It must be the maximum CONTINUOUS current, not the maximum current.