The website likely sourced its information from the website of the Belgian government, which indeed used to state the above. However, the government's information was incorrect (and has been removed). The more comprehensive sources which I have consulted (
including this one) state that King Albert I renounced the usage of German titles (not family names) in April 1921 (not 1920).
This topic has been discussed extensively (and additional sources are posted) in the "Titles of the Belgian Royal Family" thread.
"The King of the Belgians" is his official title, but the Belgian linguistic convention is indeed to use "King Filip of Belgium" when a speaker refers the king by his given name. I am not sure of the reasons, but perhaps it is because the King of the Belgians
is additionally a Prince of Belgium.