If I've got two batteries of the same capacity, one is 12V and the other is, say, 18V with a step down buck converter, will I essentially be able to power my load at 12V for longer? I know that a battery's voltage will decrease as it discharges, so it seems to me that a better way to power a load with a more stable voltage (and maybe for a longer period of time at this voltage) would be to power it with a higher voltage battery and step that voltage down. Is this true?
The other guys are right in theory or abstract. And certainly, whichever battery has higher energy storage will likely last longer.
But from a product design perspective it is a good idea to use a battery pack that is a bit higher in voltage than what you need and buck down to your desired voltage. So I think you are very much on the right track in your thinking and approach. A simple buck will be easier to find and cheaper and more efficient than a buck/boost.
Let us consider 12V. If you use a 3S lithium ion battery pack, your span will be something like 12.6-10V. If you need 12V, then you will have to buck/boost. But if you go 4S, then your span will be 16.8-13.2V. So you can use a buck. That is going to be easier and more efficient.
In low-current, low cost applications, a linear regulator may also be considered instead of a buck. Typically linear regulators can be designed with lower quiescent currents than a buck. So the overall system battery life may not be any worse when the low quiescent current is factored in. I doubt that is the case for you, though. Usually 12V applications are not low-power.
What matters in not voltage, it's energy.
The energy in a battery is quoted in Wh (watt-hours). Many still don't specify the watt-hour capacity but do quote the Ah (amp-hour) capacity. With this and the battery voltage you can calculate the Wh capacity from:
$$ Wh = V \times Ah $$.
Compare the two.
If you use a step-down converter you need to factor the efficiency of that into the calculations.
Assuming we are comparing apple with apples, ie. both batteries have the same energy capacity and power, there is no fundamental difference between them. So you are looking at secondary effects such as converter efficiency and the load's tolerance for voltage variation.
The primary advantage of a higher voltage stepped down is that the regulator can maintain constant output voltage throughout the discharge. If your load requires at least 12V for proper operation then this could extend run time significantly (if the '12V' battery drops below 12V toward the end of its discharge).
On the other hand the step-down regulator itself wastes some power, so the higher voltage setup may actually perform worse. Achieving high converter efficiency at both high and low currents is difficult. A load that needs high power for short periods (eg. DC motor) and/or spends long periods in standby (eg. telemetry transmitter) might be better powered directly from the battery.
Finally you should consider the extra bulk and weight of the regulator. To be truly equivalent the two systems should be the same size and weight, which means the higher voltage battery may have to be smaller to accommodate the regulator. A good example is a drone, where any weight increase results in lower run time. A higher voltage stepped down would suffer from extra loss in the buck regulator, plus the extra weight of the 'oversized' regulator required to handle peak motor currents.
You are forgetting that there are converters that can both step up or step down the voltage as appropriate. So you don't actually need a higher battery voltage than your load voltage. But you do need the battery to have enough energy left in it, whatever its actual voltage is relative to the desired load voltage.
A dead battery is a dead battery, no matter how high or low its voltage is. But just because a battery has a lower voltage than you need for you load doesn't mean it has no energy left inside it that can't be used to step up the voltage.
Only something to add:
Higher voltage battery has more cells in series. The total voltage with a regulator probably stands usable when the weakest of the cells is exhausted. That weak cell starts to get charged reversely when the rest of the cells still output current. The weakest cell can get serious punishment during that process. Advanced battery packs have individual cell monitoring circuits to prevent self-destructive usage like this.