Is it permissible to continue saying "I intend the purification of ablution" while washing the limbs, or should one adhere to what is customarily said when performing ablution?

The fundamental principle regarding acts of worship (ibadat) is that they are prescriptive (tawqifi); nothing of them is legislated except with evidence. Any act not brought forth by the Sharia is rejected, because an action is only accepted if it is performed with sincerity to Allah and in accordance with the Prophet’s guidance (peace be upon him). The Prophet (peace be upon him) said: "Whoever does an action which we have not commanded, it is rejected," and "Hold fast to my Sunnah and the Sunnah of the rightly guided Caliphs after me. Cling to it with your molars. Beware of novelties, for every novelty is an innovation, and every innovation is misguidance."

Uttering the intention (niyyah) aloud in acts of worship is an innovation (bid'ah) that was not transmitted from the Prophet (peace be upon him) nor from his Companions, and it indicates a deficiency in intellect and religion. A Muslim must abandon what the Prophet (peace be upon him) did not bring and steer clear of innovation in religion, and be eager to learn the matters of his religion.