A typical use of the power supply schematic in the O.P. is for powering one (1) analog circuit, which needs both positive and negative power supply rails. In principle, you can power two (2) completely independent circuits from it (+9 to 0 and 0 to -9). This would be a peculiar scheme, though, because the ground of one circuit is at 9V (plus or minus) w.r.t. ground of the other one. Still, if the circuits are in fact independent, such scheme would work. In practice, this scheme is not used in general purpose desktop power supplies.
A classic desktop power supply (like the one in the YouTube video linked in the O.P.) is of a "dual positive type". Each channel has its own independent secondary winding (or even a separate transformer) and its own rectifier. The channels can float with respect to each-other, and that allows to connect them in series.
Your calculations are correct. 354 LEDs at 20mA each is 7.08 A (354 * 0.02).
The thing to be careful about with LED strips is the voltage drop. The copper thickness on strips like the 5 meter one you have tends to be 0.5 to 1.0 oz, and the voltage drop is quite noticeable. In other words, if you supply voltage to one end of the strip (12V), the opposite end will only receive 10-11V. The instructions may indicate that you should supply power at both ends of the strip, which is a good idea. Personally, in my projects, I do not run strips longer than 2 meters with that type of copper.
Edit:
Looking at the datasheet for the LED strip, it is a 5 meter strip with 60 LEDs per meter. Each "LED" is actually a 5060 package SMD RGB assembly, which means there are three LEDs per package. The datasheet specifies the maximum If at 30mA (not 20) for each color, R, G, and B. (The characteristics are derived using a test If of 20mA, but the absolute maximum value is a worst-case scenario, which you should use in calculating power supply size.)
Unless I am reading it incorrectly, this means to light the strip at 100% brightness at white (all three colors), it will need 90mA per LED. That works out to 5.4A per meter, or 27A total.
Comments on the product page suggest that the strip only requires 6A. I suggest measuring to be sure. (Strips that I have require about 20mA per color, which is what made me think to come back and add this addendum.)
Best Answer
Calculate power consumption for each individual circuit (as if you had wattmeters in each seperate power feed) and add the numbers up. For instance: -
simulate this circuit – Schematic created using CircuitLab
You could calculate power as 20V across a 20 ohm resistor and get 20 watts BUT you don't necessarily know that currents from both supplies are the same. So, you should look at it from the perspective of two individual power supplies feeding their individual currents into a black box (and that black box might just be disposing of some current to ground).
Either way you'd still get the same result, 20 watts, for the example above.
You could just imagine it as two 10 ohm resistors each connected to 0V and, if you do the math and you would get the same result but, what if those two resistors are unequal - you are forced into summing the individual powers based on individual voltages and their individual respective currents.