Well, if you get the 6v Sullivan pump, get a "2s" battery. Any amps - but 700 mah lasts me a long time. At least a full gallon, but I haven't pushed it beyond to see how far I can go.
If getting the 12v, get a "3S" or "4S" (see below)

The "S" numbers are how many cells in the battery. LiFe are 3.3v per cell, so "2S" means 6.6 v, 4S means 13.2 v
LiPo's average about 3.7 v, so 2S = 7.4, 4S = 14.8
"3S" batteries are not common, so can be harder to find.

Hence why I use the LiFe cells. A bit safer in their own right, and even fully charged, when all batteries have higher voltage than their nominal rating, I won't be greatly exceeding what the pump was designed for.
Since 2S is cheaper, and the voltage is closer to what the pump was rated, that's why I went with a 6 volt pump.

For the Dynatron, I went with a "6S" LiFe, which puts out 19.8 volts. Lots of extra power, yet not pushing it to the max (rated for up to 24 v.) Note that there have been instances of the switch inside the starter actually welding itself "on" when using 24 volts...so no need to go super big....)
For the the regular Sullivan or the Dynatron, either flavor of "4S" would be fine, also. Physical size is the only real constraint, depending on how you decide to attach it to the starter.

Cheaper chargers generally use a 12v battery for power - car battery, etc. To plug it in using wall power you'd then need an external power supply to convert the wall power to 12v. More expensive chargers can do either. I use this one, but it is not cheap: https://hitecrcd.com/products/charge...supply/product