yes this is true. it doesnt matter how long the tank has been running with water. your cycling process starts the moment you add a source of ammonia, in your case, the day you added your betta.
the reason your ammonia is reading 0 is because you have a tiny little fish that hardly produces any ammonia. just because your readings are 0 does not mean the tank is cycled. you either need to add more goldfish, or pure ammonia, depending on what route you wanna take.
if youre gonna do a w/ fish cycle, as i mentioned, add more fish and give it about 6 weeks, thats how long a normal cycle will take. you need a good source of ammonia to start feeding the bacteria in your filters, and one little betta is not gonna cut it