This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $f$ to count as explicit, one shouldn't need to know in advance that there exists a bijection in order to believeprove that $f$ existsis a well-defined bijection. So for example if you order the elements of two sets $A$ and $B$ in some way that has nothing to do with why $|A|=|B|$, then you need to know that $|A|=|B|$ in order to conclude that the bijection that maps the $k$th element of $A$ to the $k$th element of $B$ is indeed a well-defined bijection.

I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy).

This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $f$ to count as explicit, one shouldn't need to know that there exists a bijection in order to believe that $f$ exists. So for example if you order the elements of two sets $A$ and $B$ in some way that has nothing to do with why $|A|=|B|$, then you need to know that $|A|=|B|$ in order to conclude that the bijection that maps the $k$th element of $A$ to the $k$th element of $B$ is indeed a well-defined bijection.

I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy).

This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $f$ to count as explicit, one shouldn't need to know in advance that there exists a bijection in order to prove that $f$ is a well-defined bijection. So for example if you order the elements of two sets $A$ and $B$ in some way that has nothing to do with why $|A|=|B|$, then you need to know that $|A|=|B|$ in order to conclude that the bijection that maps the $k$th element of $A$ to the $k$th element of $B$ is indeed a well-defined bijection.

I think this criterion rules out 1 and 4 (or would do if one could make it more formal, which might itself not be wholly easy).