Question regarding notation of a language decidability $1.\: A_{DFA} = \{\langle B, w \rangle \mid B \text{ is a } DFA \text{ that accepts input string } w \}$
$2.\:A_{DFA} = \{\langle B \rangle \mid B \text{ is a } DFA \text{ that accepts input string } w \}$
I know how to proof that 1 is decidable by constructing a machine that always halts and accepts whenever $B$ accepts, otherwise rejects. what's the difference between 1 and 2. I find that it doesn't make sense to not include $w$ inside the encoding brackets but I have seen this notation in other places. are they both the same?
The languages are not the same.

In the first one, $w$ is a part of the input.
In the second one, $w$ is fixed beforehand, and the language has to depend on what you fix it to be.