There is no specific concept of "length", "width" or "height" with these measurements. The longest measurement of your item must be no more than the longest allowed number, the 2nd longest measurement no longer than the 2nd longest allowed number, etc.
After all - if your bag is 50cm high, and you turn it on it's side, it is still 50cm high or is it now 50cm long?
As for your question of what is the maximum allowed common carry on baggage size, it would be the smallest length in each dimension across the three carriers. Thus, 55cm in one dimension (Emirates/Turkish), ~36cm in the second dimension (Copa/Bahamas*), and 20cm in the third dimension (Emirates), with a weight of no more than 7kg (Emirates).
How strictly these measurements are enforced varies depending on the airline, the airport, the specific staff involved, and sometimes how full the plane is. It's possible you will be allowed travel with an item larger then the above measurements, but there is no guarantee.
(* 14" = 35.56cm)