This is actually a rather frustrating example. Thus far this unit, we've been marked wrong for putting in phrases like, "My weird dog" instead of "My dog is weird" and "My amazing daughter" versus "My daughter is amazing". So then to have this one thrown in and zero explanation as to why one is right and the others all wrong doesn't help achieve any sort of understanding goals.
I think this one should be removed until later in the course when I assume (and hope) will learn the difference and when these scenarios are okay.
To answer this, you should be able to say if the adjective is "definite" or "indefinite". Let's have a look at your two examples :
My weird dog -> كَلْبِي ٱلْغَرِيبُ (kalbii l-ghariibu); My amazing daughter -> بِنْتِي ٱلْمُمْتَازَةُ (bintii l-mumtaazatu)
Note that the adjectives begin with "الـ" (el) , the definite article. It means that you know, while speaking about the subject, that he has this quality. You know that my dog is weird, that my daughter is amazing,
I want the big book -> أُرِيدُ ٱلْكِتَابَ ٱلْكَبِيرَ (2riidu l-kitaaba l-kabiira)
Here again, you "know" the book (that one on the table, the one I see, etc...), and you know that it is big. Then you say THE big book.
Note that when it is the case, the word and its adjective bear the same vowel at their ends. In my last example, "el-kitaaba l-kabiira", you notice that both words start with "el-" and end with "a", (it is this way in the accusative case for singular).
My dog is weird -> كَلْبِي غَرِيبٌ (kalbii ghariibun) ; My daughter is amazing -> بِنْتِي مُمْتَازَةٌ (bintii mumtaazatun)
Here, you don't know more about neither my dog nor my daughter. the adjective comes here to give the information. And to give it, you need the use of verbs like to be, to seem, to look, etc... In the examples above, it is the verb "to be", which has no équivalent in arabic ( in russian, too...). In that case, focus once more on the beginning and the ending of the words : Here, غَرِيبٌ and مُمْتَازَةٌ don't start with "el-" and end with a "tanwiin Damma", pronounced "-un". If there is a "tanwiin", then the word is indefinite. Hence, you will never find a word like الْغَرِيبٌ (definite with "el-", then indefinite with a "-un" or another tanwiin).
I want a big book -> أُرِيدُ كِتَابًا كَبِيرًا
You don't speak about a specific book (indefinite, hence the use of "a" in english, and a tanwiin with the absence of "el-" in arabic).
One last example : the book is big -> ٱلْكِتَابُ كَبِيرٌ (el-kitaabu kabiirun)
Youn know the book. it is definite with "el-". You give the information that the book is big. Note that the adjective doesn't start with "el-" and ends with a tanwiin ("-un", which is nominative).
Overall, the definite words are those with "el-", the proper nouns, and the words starting an "iDaafa" (genitive case construction), like كَلْبِي (kalbii, "THE dog of me") and بِنْتِي (bintii, "THE daughter of me").
Do tell me if this helps, or if you need more explanations.
NB : a shame that DL doesn't teache this vowel ending thing. The sentences can have multiple meanings if you don't pay attention to this.