Your controller is probably a 20 amp controller, which means that it will try to draw a maximum of 20 amps from the battery when you're going slowly up a hill or are accelerating from a start. Amps is like water flowing down a pipe. The more amps, the more water is flowing.
Rather than physical size of battery, we refer to how much electricity is in the battery in amp-hours (aH). A 10aH battery can give 10 amps for one hour or 5 amps for 2 hours, so imagine water flowing out of a tank, the faster it comes out (amps) the quicker the tank (battery) becomes empty.
Size matters. The bigger the battery in aH, the more current it can give, and the general rule is that the numbers are the same, so a 10aH battery can give 10amps continuously, and a 20aH one can give 20amps.
Your controller is probably 20 amps, so if your journey has a very long hill, where you'd be using 20 amps for a prolonged period, you'd need a 20aH battery; however it's OK to exceed the batteries rating for short times, so you can get away with a smaller battery. The lower you go, the more problems you'd have. A 9aH battery can only give 18 amps max for a short time and 9 amps continuously, so if you try and draw 20 amps from it, the chemistry can't keep up, so the voltage drops until it reaches 29v, when the LVC kicks in and switches it off. Leave it for a few seconds and the chemistry will catch up and it'll come back on. Many people with the 9aH bottle batteries experience this effect because their controllers are rated at 15 amps.
Some batteries, like the white 10aH one that I indicated in a previous post have special chemistry that allows a higher discharge rate. It can give 30 amps max, so it's equivalent to a 15aH normal battery, but it's much heavier and the same weight as a 15aH one, which cancels out its main advantage.
Even though your controller is probably a 20 amp one, most of the time, you'll be using a lot less current, so you don't really need a 20aH battery. The lower you go, the more likely you'll get cut-outs and the more you shorten the life of the battery. There's no exact answer to the minimum size from experience I would say that 12 aH would be the minimum, 15aH (or equivalent) would be better, but heavier and more expensive.
The 12aH battery that I showed you previously has a Kettle plug on it, the same as what you already have, so that particular one would be the most logical choice. The other one has a 4 pin connector, so you have to make a joining wire with the 4 pin on one end and kettle plug on the other, or cut the kettle socket off your controller, extend the wires and solder on the 4 pin connector that they give you with the battery. Unfortunately, they placed the 4 pin connector at the far end of the battery (daft) which is why you have to extend the wires. The 12aH has it at the near end, so just plug in.
Whenever you get a new battery or charger, you should always check with a meter, which way round the plus and minus wires are, otherwise you'll hear something go pop and see smoke - often lots of it!
