I wouldn't think of a kanji as a word, it's a character with has a meaning with japanese word(s) attached to it. So, there is no definitive way to say a kanji. There is many correct ways to say most kanji, however when reading, there is only one correct way to say it. A kanji usually has more than one reading, which is used depends on the situation. If you know when to use which reading, it's usually straightforward. Onyomi usually are only used in compound words with more than one kanji. Therefore, とう is not correct. Kunyomi are used usually when the kanji is by itself. あずま refers to Eastern Japan, not used too much. とう is the onyomi, it has no meaning by itself. ひがし means "East" and is what this kanji is usually read as. If you see 東 by itself, it would be read as ひがし.

Think of "that was close!" vs "close the door!" or "I read books" vs "I read that book." Each time you knew how to pronounce close and read, right? Kanji words work exactly like this. The pronunciation changes based on context, which Kanji or hiragana it's paired with, etc. So it's best to think less about how many pronunciations a single Kanji has and focus more on learning vocabulary and how they're used in sentences.

don't learn "just Kanji", learn "words with Kanji"