When someone says "current", they mean the combination of volts and amps, right?

I don't. Current is measured in amps. Voltage over a resistance causes a current to flow through the resistance according to Ohms law: U=RxI (or I=U/R) (where U = voltage, I = current, R = resistance)

In all these analogies, you guys keep saying "current", and I'm looking for a way to grasp what the two different halves of "current" represents. I haven't yet seen anything in the water analogies that shows me this. To me, it's all the same. More water= More pressure, even in the soda bottle analogy. So in that analogy, if the height of the water represents volts, then what represents amps?

The water flow through the opening (volume per time unit), and the size of the opening represents the inverse of the resistance.

A taller water reservoir corresponds to higher voltage.
Connect a water reservoir to a straw and you get a fairly small flow at some speed. The straw is high resistance. The flow is the current. Connect a 3' pipe and you get a much larger flow from the same height water reservoir and at the same speed... The larger pipe <-> less resistance.
If you increase the pressure enough (water pillar) you can get the same flow through the straw (at a much much higher exit speed) as you got out of the 3' pipe above

Effectively, current (amps) can't exist without voltage: you need a voltage difference over a resistance to cause current to flow. Or one can see it like current passing through a resistance giving rise to voltage. In the water analogy, water flow passing through a small/rough pipe would cause a pressure drop...

/Michael