Figured it out. Velocidrone distinguishes between Custom tracks and Leaderboard tracks.
Top 200 speed records for the official Leaderboard tracks are recorded by the Velocidrone server and displayed on its website as well as in the Leaderboard dialog inside a track.
For the Custom tracks, however, the best speeds can be only seen in the Nemesis mode (which is the third game mode, in addition to Multiplayer and Single Player).
Velocidrone's official channel also has videos explaining leaderboards and Nemesis mode.
One important thing to notice is that for your best times to be uploaded to the server and displayed on the Leaderboards you have to press the upload button after completing a race or toggle Auto Leaderboard Time Upload to Yes in the Main Settings.