I think you are still confusing the max current a battery can provide with what it will provide to a given load (if able).
Most power sources, including batteries act as voltage sources, not current sources. That is they stay at a certain voltage and as load is added the current flowing increases to keep the voltage where it was, up to a point. This point is the current rating you are talking about with 7 and 70 amps.
A current source would try to push the same current all the time and therefore a greater load would require a greater voltage between the poles of the source. Given a voltage source, like you have, a current limiter doesn't make a ton of sense, as you presumably want you solenoid to recieve the max amount of current it can draw so it operates as designed.
The solenoid will try to draw a certain amount of current at a given voltage, the battery may or may not be up to providing that current. If the battery is not up to providing the current the voltage will droop, you may have seen this if you have ever shorted a circuit while measuring what should be a non-zero voltage, the voltage drops and something will start getting hot.
When you used battery capable of sourcing more current, the solenoid will draw more current. There is a difference between the total amount of power a battery will over a charge cycle, measured in Amp hours, and the amount that it can output at any given time.