I can see why you would be confused. The Oxford definition is, no doubt, correct; but it is pretty complicated for everyday purposes. Let me explain it the way I do when speaking to American school children.
Whenever you are trying to decide what part of speech a word (or phrase) is, look at what job the word/phrase is doing. Adjectives and adverbs both have a job of modifying other words. Basically, that means that the modifying word/phrase gives the reader/listener more information than the modified word alone would. Adjectives modify nouns; and adverbs modify verbs, adjectives, & other adverbs. (Also, for some reason, it is typically more difficult for students to identify adverbs. That is why I teach them to think about adverbs along with adjectives.)
In English, there are words that complete verbs rather than describe them. In most cases the completing word is a preposition. For example, all of these verbs have different meanings: get, get into, get over, get along, get about, get going, get underway. However, "get" is the only case where the meaning is clear without the second word.
So, going back to your sentence: "I'm glad" would be a complete sentence — a subject and verb. Your reader/listener could find out more about "am" but s/he understands fully what the verb means. The phrase "that we won the game" tells your reader/listener more about "glad" rather than completing the verb "am" so it modifies an adjective and thus is an adverbial phrase.