I am not particularily familiar with the connotation of the term second-order-thinking, but to me it implies the somewhat mathematical meaning of it. The first order might be some value (e.g. a voltage) and the second order might be the rate with which that voltage changes and the third the rate with which this rate changes etc.
To me this implies that thinking on a higher order is somewhat about stepping back and viewing the construct from a meta position. E.g. instead of thinking about how to address an imidiate problem by applying a bandaid, one order higher would be to think about the root causes and deal with the root of the problem. Even one order higher would be to think about which culture or systems of thinking allow for the good solving of such problems etc.
To me this implies that thinking on a higher order is somewhat about stepping back and viewing the construct from a meta position. E.g. instead of thinking about how to address an imidiate problem by applying a bandaid, one order higher would be to think about the root causes and deal with the root of the problem. Even one order higher would be to think about which culture or systems of thinking allow for the good solving of such problems etc.