I feel that you must go to the basics of the Electrical Engineering (or Physics - Electromagnetism)
In any Electrical generator what is generated is Electro-motive Force (Refer faraday's law/maxwell's Equations - any time varying magnetic field generates an Electric Field)
As you must be aware that the electric field is created by charge and is measured in Volts /meter. The volts (or the voltage if you like) are the measure of the electric field.
As you connect the two ends of the generator with a conducting media, the current flows through it, limited by the imdedance of the connection (Resistance only in DC, Inductance and capacitance added in AC systems)
So current is the effect and the Electric Field Potential is the cause.
I have talked about Electrical Generator, but in any of the active devices, what is created is the EMF (Electro motive force) - Cells (batteries) create them by concentrating electrons on one electrode and strips them out of the other,. Solar Cells do the same - direct the electrons to one side.
And this concentration of charges are what create the Electric Field- you must know the equation (and that can be translated into Electric Potential or Volts)
The best analogy (often used to be told in our classes) - Voltage is the pressure , EMF (Electromotive Force) is the one at the pumping point and hence is constant. The local Pressures (Volts) vary along the pipe line affected by pressure drops (between that point and the source). The pipe line resistance (frictions etc) limit the flow (current).
What you do in case you need a constant flow pipe line? You create one accumulator (tank) and have sufficient high pressure inside it. Connect the load through an orifice. Flow through orifice would be dependant on (inlet pressure - load back pressure) If this is high enough, then the flow will be practically constant. Another way may be to use constant flow (Positive Displacement) pumps.
In electrica also we have ways of making flow constant but what is finally created is the Electric Field and Voltage is measure of that.
Yah, this depends on the design of the power supply. If you have a constant current design in mind, the voltage is dependent on the load--including leakage and opens, i.e., infinite loads.
Constant voltage supplies are more common...like wall bugs, PC's, etc. Current varies dependent on the number of loads. Your home is a great example of a load for constant voltage, variable current.
The perspective of; 1. Where is the source and 2. Where is the load makes a big difference. Do you consider the power supply in your computer a source? What about the wall plug that feeds it? What about your load center in your home? What about the transformer, substation, etc. See where that's going?
A lot would also depend on whether the constant was designed for AC or DC loads.
Constant voltage sources are the more common model people naturally think of because a battery closely resembles an ideal voltage source. Few things here on Earth closely resemble an ideal current source. Yes, many important circuits can be engineered to work as a current source in expected conditions. But none of these circuits are as easy and simple to produce as this:
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"Don't disturb my circles." translation of Archimedes last words