The different colours are indeed bronze, silver and gold, just like you listed.
The difference between the colours is that silver and gold are more difficult to get, because they require you to do the same thing more often. If you look at the description column of the medals page you listed, the exact requirements are listed.
For example, for the standard medal, the requirements read (important part bolded):
standard medal requirements wrote:Awarded to members that defeated X unique opponents in Standard games. Bronze=20, Silver=100, Gold=400.
So once you have defeated 20 unique opponents in standard games, you'll receive the bronze medal. When you have defeated 100, this gets upgraded to a silver medal, and at 400 unique opponents defeated it becomes gold.