18 kt gold is generically harder than 24 kt gold. Both the site you linked and another site agree on that. Pure gold really is quite soft, and when you add things to it, it will get harder.
Now, one key to your problem with understanding the hardness of gold and its alloys is that multiple different alloys can be called "18 kt gold". Different alloys will have different properties and hardness. Interestingly, there is a titanium/gold alloy that is ~4× harder than titanium normally is. The paper - Science Advances 2016, 2 (7), e1600319 - on it looks interesting.
Back to the topic on hand, note in the table below that there are different values for hardness for each carat of gold based on how it has been physically worked and how it has been alloyed. Also note that all the values listed for 18 kt gold are significantly above that of 24 kt gold.
Source: Total Materia; Gold and Gold Alloys; Table 3: Mechanical Properties of Typical Gold Alloys
It might be worth a separate question of why 9 kt gold might be listed as softer than 18 kt gold, if it has more in the way of impurities.