Grammatical terms are not always used consistently in different sources. In general, the term noun modifier is usually a broad term, often encompassing adjectives, nouns used to modify other nouns, verbs used as modifiers (usually past or present participles), and even phrases and clauses that modify a noun.
- Adjective: I took the big tire to the garage.
- Noun: I took the truck tire to the garage.
- Verb: I took the damaged tire to the garage.
- Verb: I took the spinning tire to the garage.
- Prepositional phrase: I took the tire under my car to the garage.
- Phrase with participle: I took the tire damaged by the nail to the garage.
- Clause: I took the tire, which had been given to me by Uncle Joe after the war in exchange for two bottles of sarsaparilla and my solemn promise not to tell my parents about it, to the garage.
Depending on the grammatical guide you're using, any or all of these things could be called "noun modifiers."
In contrast, an adjective is a specific part of speech. Some grammar guides would allow the use of "adjective" to refer to things like your example of damaged tire, but others might limit the term to words that are not derived from other parts of speech, like big, small, happy, funny, etc., while referring to other words functioning as adjectives to be "participial adjectives" or within the broader category of "noun modifiers." There's no question that in the second sentence the word is functioning as an adjective, whether one calls it that or not.
The problem with limiting the group of "adjectives" too much is that some adjectival forms created from verbs (and occasionally other parts of speech) acquire specific meanings and associations in their adjectival form. For example, in the phrases fallen angel or fallen woman, the word fallen is not merely a past participle of to fall, but has acquired specific connotations. Other verbal adjectives are based on archaic root words that are rarely or never used in their original form. For example, dilapidated is a common adjective, but the verb dilapidate has fallen out of a use for the most part, so has dilapidated become the primary word as an adjective now? Dictionaries often differ in their choices about when to list a derived word as a separate adjective.
All of these cases make it rather difficult to draw a firm line between adjective versus other types of modifiers. Nevertheless, if I were required to use a book that created such a distinction, and I had to explain the concept to a class of students, I'd start with the idea of adjectives as words not derived from forms of other parts of speech, as in my list above including big and so forth.
It's not a perfect explanation, but few grammatical terms are ever perfectly precise.
As a side note, there's a bit of a question about whether the word damaged in the first sentence, "My tire was damaged," is necessarily a noun modifier. Depending on the broader grammatical context, it could also be the verb in a past continuous tense. In the context "my tire was damaged yesterday by a nail," damaged is part of the verb. Compare that with the context: "I ran over a nail last month. Yesterday my tire was damaged; today it is repaired." In that case, damaged is likely a noun modifier. With the limited context given in your two sentences, it isn't even clear that damaged in the first sentence is even used in an adjective-like sense.