Which of the following is recursive
No, your reasoning is not correct, but I think you are very close to a correct answer.

A Turing machine in 2000 steps can only read the first 2000 symbols in the input tape, but is not forced to reject a longer input. In fact, it can accept it. So, the TM can process longer inputs, but its behavior is determined by the first 2000 symbols on the tape.

If $x$ and $y$ are words which are identical on their first 2000 symbols, then they are both accepted/rejected/neither in 2000 steps.

Because of this, we can partition the possible inputs according to their first 2000 symbols. For each of the finitely many partitions, we can extract any element and test for that one, only.

So, our decision procedure is the one given by the OP. The argument for correctness is a bit different, though.