Yes, it is a myth. Same as drinking beer dehydrates you. Yes, drinking either will make you pee, duh. Taking a caffeine pill before a ride will not make you pee. QED.

The main effect of caffeine for a bike rider is that it spares glycogen. The caffeinated rider derives a larger portion of energy from fat stores at normal riding intensities. It's a good thing. IME taking caffeine at the start will not make me more tired at the end. IOW it's not necessary to wait until one is tired to derive benefit from caffeine. In fact, the opposite may well be the case.

In this same myth-busting vein, it has now been well established that the best hydration one can do after a ride is to drink one beer. On a recent 3-week bike tour in the Czech Republic, I drank 1/2 liter of beer every riding day with lunch and another after riding. I would say it had a positive effect, much like any sports drink would. Czech lagers are not strong, usually about 4.4%. Although I did not notice any performance-depressing effect of the noon beer, it did have a psychologically depressant effect, a well-known effect of alcohol. But it tasted so good I drank it anyway.