I am making an indoor fountain with an old 800 L/H AIO pump. Will the pump run full speed without the signal, like fans? if so, can I hook a potentiometer to the 12v adapter to adjust the height of the stream?
An actual PWM fan WILL run with no PWM signal, and its speed WILL be controlled by reducing the voltage to it. However, virtually all AIO pumps are designed NOT to do things that way. Most actually are designed and wired just like the older 3-pin fans with NO special chip to use the PWM signal for speed control.So, when you plug one of those into a header that actually IS putting out the standard signals for a 4-pin PWM fan, it does the same as a 3-pin fan: it runs full speed all the time because that is what is supplied from the fan header on Pin #2 - fixed 12 VDC at all times.
So, assuming your old pump from a AIO system is the common style, it really will act exactly like a 3-pin fan would, and its speed can be controlled by reducing its supply voltage. But you ought to take into account start-up requirements and minimum voltage. A fan (or pump) needs a certain minimum voltage to just START when it is not running. This is even more important for a PUMP that must start up "under load" filled with water and trying to force it through. In a computer, the most common practise for FANS (applies to any pump, too) is always to START at full 12 VDC to be sure it does start, THEN in a few seconds reduce the voltage to what is required at that time. You could do this if you have a "pot" as a speed control simply by YOUR habit of always starting by turning it to full speed, then turning down to your preferred setting.
Next point: a common 12 V fan actually will stall if its supply voltage is reduced too low - below 5 V is typical stall Voltage. Once is stalls, it can never re-start until it supply voltage is increased significantly to above the minimum needed for starting. For a PUMP under load, that stall Voltage may be a bit higher. Recognize, too, that a STALLED fan (or pump) that still has a voltage supplied to it WILL be drawing current and heating up its windings, but has no way to cool itself. So a STALLED pump is something to avoid. You should rig some way of preventing turning down your speed control below some limiting position.
Here's where I get picky. We use the term "Potentiometer" for a variable resistor consisting of a resistance strip with a supply voltage across its entire length, and a moving contact that can travel the entire distance along the strip. But that device can be used two ways. If you connect it so the OUTPUT of this device is the voltage between the moving contact and one end (say, the end connected to the (-) power supply line), then the moving contact provides a Voltage or Potential according to its position. So this IS using it as a Potentiometer. On the other hand, if you connect that device just in series with a load - that is, connect ONE of the power supply lines (say, the (+) source) to one end of the unit, and an output connection wire to the moving contact that feeds to the load device, and then the (-) power line goes to the other terminal of that load, then the device is a Variable Resistor (also called a Rheostat) in series with the load. The actual Voltage arriving at the load depends on the current being drawn in the total circuit, because that current produces one Voltage drop across the resistor and another remaining Voltage drop across the load. VERY often for varying a voltage supplied to a small load (and hence its speed) it is this SECOND type of circuit used, in which the device technically is not functioning as a Potentioneter. BUT we still use that name for that device.