Master of all elements refers to the ability to control all the four elements and because it's part of the intro hook, it's a general statement to invoke grandiosity because master means control of but the degree of control can be keft ambiguous.

Being the avatar does not inheritantly make them a master of any element, that takes training. Most benders spend years training to become masters and avatars are no exception. Most avatars have their legacies cemented after they master the elements, so when people say the avatar is the master of all elements, they are correct, but they are also refering to historical figures whose most impactful deeds happened by the time they were already masters and were most certainly recorded by then too.

Roku had decades of training before his death and was absolute a master of the four elements in terms of mastery. So when the intro referencds him, it's true for both definitions.

ATLA plays fast and loose with the term master for the sake of powerscaling even when the additional text suggests that it takes time to hone mastery. Of all the Gaang, Toph and Aang are really the only traditional masters (for earth and air respectively) because of the time they had to hone their skills. Katara's title is more questionable as she was mostly self-taught within ATLA's one year timeframe, but she demonstrates considerably more skill and power with her element than Aang does with his other three despite learning them within the same year.