Electronic – Can a 12V power supply provide more than the rated current at lower voltages

The LED does not affect the load of the voltage regulator. It affects the supply. The voltage regulator's load is the 9V device. The only way it will cause a problem for the regulator is if its current draw triggers a voltage drop below the point that the regulator is able to maintain 9V.

The main question about your circuit is: is the incoming 12 volts regulated? Or is it just the nominal voltage on a transformer device such as a wall adapter? If 12V is the maximum rating of the LED device, but the supply is unregulated, you may want an additional 7812 regulator there.

(What kind of LED is it; and does it have a built-in resistance for current limiting?)

Current flows in loops (always) and your power supply actually has two loops that matter for this.

Simplifying slightly there is a primary side loop, Live -> Psu -> Neutral (completed by the power companies equipment), and a secondary side loop made up of the 12V wiring, the 12V return and the load.

In both loops the same amount of charge flows into and out of the PSU on its respective loop, so on the DC side it is 41 coulombs/second (at full load), and on the AC side it is an average of around 4 coulombs/second (Actually closer to 5 or 6 probably for various reasons, also on the ac side this is slightly simplified).

In both loops the same amount of charge exits the supply on one wire as enters it on the other, energy is converted by moving the charge thru a potential difference and it is energy that is transferred between the primary and secondary circuits, not charge.

Note that because the same amount of charge exits on one wire of the loop as enters on the other (to a very good approximation), there is little net charge built up in the power supply, it this condition did not hold the charge on the supply would build until something (quickly, and probably explosively) failed.

Regards, Dan.