I just assumed that this meant the Crown and/or the British Government was no longer giving awards (my first thought was knighthoods, but I think we can still get those - not sure about other awards) to Commonwealth citizens who weren't British,

The British Government hasn't awarded medals to Commonwealth citizens for decades - the nations involved made the awards themselves. Most of those countries have now abandoned the practice, substituting their own honours instead. There are still a few awards that are the personal gift of the Sovereign, but in many Commonwealth countries, even these could only be accepted by the intended recipient with the express permission of their government.