Difference between current and power
Answering all your questions properly will take some effort. Let us consider the case where we have a voltage supply with a fixed output voltage. This is the most common case for off-the-shelf products.

AC appliances mostly use a fixed voltage (depending on country). Since the voltage is fixed and known, I can calculate power if I know current, or if I know current, I can calculate power using P = I * V. This is probably why you consider them to be kind of redundant or closely related.

Now let's consider a different case. Suppose I have a 3.7V battery. I want to use it to power a 5V circuit that uses 100mA. In order to do that, I have to raise the voltage to 5V (using a boost converter). Now, power must be considered separately from current. The power required by the circuit is 5 * 0.1 = 500 mW. Due to conservation of energy, I will need at least 500mW from the battery. In reality, I will probably need around 600 mW, due to less than perfect conversion efficiency of the boost converter. So that is around 3.7V / 0.6W = 162mA.

Power supplies may have various things specified, depending on what they are used for. Lab power supplies usually specify max current and max voltage. Adaptors for laptops probably specify maximum input power consumption, output voltage (fixed) and maximum output current.

When driving LED's, you usually start with the current you wish to run through the LED. The voltage does not vary too much with current. But when current and voltage are both known, the power can be calculated trivially (P = V * I). But in fact, white LED's intended for illumination are often rated by power. If you are buying LED's and don't have a model number or data sheet, you should consider getting your LED's from another source. It is true that 20ma is a common max current for LED indicators. But depending on the use, sometimes you can use much lower current (like 1 or 2 mA) especially for red LED's. LED's for illumination can use much higher currents than 20 mA.

A final comment. Sometimes your power supply voltage is higher than the required input voltage of your circuit. You can use a linear regulator to reduce the voltage. In this case, the current will be the same for both circuits. The linear regulator simply converts the extra power to heat. But you can also use a buck converter. A buck converter will convert the higher voltage to a lower voltage somewhat more efficiently. Typical values are 80% to 90% efficiency. This means that the buck converter will produce less heat than a linear regulator.

I skipped over some details, because I don't think you are ready for them. Perhaps some people will comment on this.