In الْقَاضِي, the word ends in yāʾ preceded by the kasra on ḍād: ضِ + ي. This is the reason it is classed as manqūṣ.
The definite article الْـ makes the noun definite. It does not cause the noun to be manqūṣ; the final sequence does.
This is an instructional analysis of the book’s example.