Correspondence of an ensemble

Let $mathcal{E} = { p_i, psi_i }$ be an ensemble of quantum states where $p$ is a probability distribution, what exactly is its physical and mathematical correspondence?

Is it a mixed state where $rho = sum_i p_i |psi_i rangle langle psi_i|$,

or a "flagged" state where $rho = sum_i p_i |i rangle langle i| otimes |psi_i rangle langle psi_i|$,

or a product state $bigotimes_i |psi_i rangle langle psi_i|^{otimes p_i N}$ for some large number $N$, of identical independent systems?

The above are all viable structures $mathcal{E}$ can be mapped to, sometimes it feels as though people can be referring to any one of these cases, but which one is it? Maybe this is common knowledge, but it seems to be rarely spoken of.