Well my inititial thought was the latter, otherwise what would there be to stop you doing, for example, HKG-SIN via LHR?

But you are correct that the AsiaMiles website does say that your lower interpretation is correct, and that you are allowed stopovers in LHR.

So can anyone shed any further light on this?

Even if the rules say that you must take the most direct (= shortest or least number of flights, or what?) available routing that is still going to give lots of routings like: DXB-BAH routed as DXB-LHR-BAH-HKG-DXB for not very many miles at all...
(after CX stops the DXB-BAH hop shortly)