I prefer to think of it in a more fundamentally accurate and perhaps more easily understood way.
The voltage of the power source and the resistance of the circuit correspond with how much current moves through the circuit. Adding more stuff onto the circuit in a way that reduces the total resistance at a given voltage allows more current through it which, if too much, can damage the circuit and/or the power source.
The power source (in this case a battery pack?) doesn't determine the total resistance of the circuit and therefor doesn't determine how much current moves through it. (Battery packs also have internal resistance which can limit how much current can potentially move through the circuit, but, when properly used, this is typically a lot lower than the total resistance in the circuit.)
The current ratings of battery packs are to show how much current they can safely produce for their own "health," (but not necessarily the "health" of the rest of the circuit and its components) and they aren't a measurement of how much current they actually will produce. It is also possible to "drain" more current from battery packs than is "healthy" for them.
Anyway, to make a long story short: Yes, you don't want too much current going through your Arduino, and you should be careful to prevent that from happening, but the power source (given that it's at an appropriate voltage level) won't "push" more current through the circuit just because it might be potentially capable of supplying/producing more current than another power source.
For most motors you'll want to use a relay of some sort, or a similar solution to prevent too much current from moving through the Arduino.