They're called articles; "a" (used in front of a noun that begins with a consonant) and "an" (used in front of a noun the begins with a vowel) are known as indefinite articles while "the" is the definite article
You're correct that the indefinite article is used to refer to something non-specific, while the definite article is used to refer to a particular entity. So
Would indeed simply be referring to any random watermelon, and
Would refer to a specific watermelon, probably one that was already being discussed or had been discussed recently, so as to provide context.
As you said you were struggling to find a use for them, here's another example:
A book is fun to read.
Here I'm simply saying that reading books is fun, and not talking about any particular book.
The book is fun to read.
Here, though, I'm talking about a specific book. I'm only saying that book is fun to read, not any others.
Note that articles exist to define nouns as specific or non-specific, and are different from adjectives because they don't describe the noun itself.