The current clamp in the picture will read the sum of the two 30A sides, and could read 60A before the breaker starts getting into its trip range.
If a neutral is present in the circuit, the two sides could have different current; but only if current is flowing to the neutral. This would require a 3 wire circuit (two hots and a neutral), not a 2 wire 240V circuit (two hots). Or, as you said, you could have two completely separate circuits on opposite phases, and are looking for the totalized usage of them together. But if you are feeding a pure 240V two wire load, then the current on both legs must match (or something is broken).
Your confusion about the 'maximum 60A' issue is a matter of naming convention. A breaker rating is based on the current per pole, not some concept of 'total current' (total current is not well defined in the industry). A 30A two pole breaker is in many ways two 30A single pole breakers side by side, and can either supply two separate 30A 120V loads _or_ a single 30A 240V load.
-Jon