Need help understanding amperage ratings for panels and sub panels
Part 1:

100% correct.

Part 2:

The way I think about it is this:


A 60A two-pole breaker - e.g., like you have in a main panel for the feeder to a subpanel - is not exactly 60A @ 240V. Rather, it is 60A @ 120V x 2. Each pole can, in theory, trip independently, but with 240V loads they would always trip together. With unbalanced 120V loads - e.g., 4 x 20A all on one leg - you can indeed end up with one leg overloaded and trip even though the other side is OK.


Remember, you are not limiting "how much power can go through the cable", you are limiting "how much current can go through each individual wire". Each wire is limited to 60A of current, with the total power, in this example, either 60A x 120V per leg or 60A x 120V x 2 for the two 120V circuits or 60A x 240V for a 240V circuit. All of which in the end are exactly the same.

Your calculations and interpretations are basically correct, it just gets a little confusing along the way.

This all relates, of course, to two key things:


As much as reasonably possible, 120V loads should be split between the two legs so that you can make the best use of the available circuits/breakers/etc. There is another small advantage, though it doesn't make much real difference - the neutral current is the difference between the hots. If you have 50A x 120V then your neutral carries 50A as well. If you have 25A x 120V x 2 then your neutral does nothing at all!
Panels and subpanels can be oversubscribed. In your example, a 60A panel could easily have 8 x 20A - 4 x 20A on each leg (and actually quite a bit more than that). That means that in theory if you plugged full power devices into everything - heaters on every circuit all on at the same time - then you could easily trip that subpanel breaker (and the same goes for the 100A main). But sizing is based on realistic expectations - and there are formulas and guidelines to follow. But some simple examples:


If you have air conditioning and electric resistance heat then it is a pretty safe bet that you won't run them at the same time, so only one (the larger) really counts.
You may have a bunch of 15A or 20A receptacle circuits in various rooms. Except for the kitchen (where you could have toaster, coffee maker, etc. all at once) most of the time only a few receptacles are in use at a time and most of those will typically be lighting (very low power with LEDs, but even incandescent might be only a few hundred watts on a 15A circuit), cell phone chargers, TVs, etc. - which might use anywhere from a few watts to a few hundred watts each. The big exception is someone with a workshop full of power tools, but most people don't have that. So a 15A or 20A circuit might have 6 or more receptacles - and that's OK as long as you don't use them all for high-power devices at the same time.
Lighting circuits are typically 15A or 20A. But even in the past (incandescent), the amount of light needed was far less than the circuit capacity. Today with fluorescent or LED a lighting circuit might use 100W or less for a large room.



So adding up the total capacity of all the circuits is important but does not directly determine the subpanel or main panel size.

In addition, as I have learned from the pros here, while the breaker in the main panel for the feeder to your subpanel is sized to match the wire (and the wire is therefore sized to fit your actual needs since larger costs more), the main breaker in your subpanel functions as a convenience shutoff switch and therefore does not need to be the same size. Going with your original example, the subpanel could have a 60A breaker in the main panel feeding it but in the subpanel itself the main breaker could be 100A (or more!) and that's OK because the ultimate protection will be provided by the 60A breaker in the main panel. If you use 60A in both then there is no guarantee which breaker will trip first or if they will both trip at the same time.