Confusion about the relation between POVMs and projective measurements

I'm a little confused about the terminology of measurement.

So say that we have the single qubit state $$|\phi \rangle=c_0|0\rangle+c_1|1\rangle$$.

If we perform the projective measurement $$P_0=|0\rangle\langle 0|$$. Then we say that the probability of obtaining the measurement result $$|0\rangle$$ is $$\langle \phi|P_0|\phi \rangle$$. So in this case we're talking about a possible state that can occur in the collapse of the wavefunction of $$|\phi\rangle$$.

In the context of P.O.V.M. measurement where we assume it's not a projective measurement, and we label the measurement operators $$E_i$$ then we say that if the result of the measurement is $$E_i$$ then we assume that the state sent by Alice was state-x. The probability of the measurement result $$\langle\phi_j| E_i |\phi_j\rangle$$.

So, in this case, it seems as though we're talking about the measurement of an operator instead , but if $$E$$ can be projective operator and in the case of a projective operator we're measuring the probability of the state collapsing into some state, then why does it seem like that's not what's happening in the more general case ?

POVMs are more general than projective measurements. Thus, every projective measurement is also a POVM, by choosing $$E_i=P_i$$.
In the case of projective measurements, we have a set of projectors $$\{P_i\}$$ satisfying the completeness relation $$\sum_iP_i=I.$$ Note that this also means they satisfy $$\sum_iP_i^\dagger P_i$$, which I would argue is more relevant. If you have a state $$|\phi\rangle$$, then with probability $$p_i=\langle\phi|P_i^\dagger P_i|\phi\rangle=\langle\phi|P_i|\phi\rangle$$, you get outcome $$i$$. If you get outcome $$i$$, the state after measurement is $$P_i|\phi\rangle/\sqrt{p_i}$$.
For a general measurement, you have operators $$\{M_i\}$$ satisfying a completeness relation $$\sum_iM_i^\dagger M_i=I$$ If you have a state $$|\phi\rangle$$, then with probability $$p_i=\langle\phi|M_i^\dagger M_i|\phi\rangle$$, you get outcome $$i$$. If you get outcome $$i$$, the state after measurement is $$M_i|\phi\rangle/\sqrt{p_i}$$. So you can see how projective measurements are a special case of general measurements.
POVMs use a slightly different formalism. They make the assumption that you are only interested in the probability of measurement outcomes, and not the state after measurement. So, they choose to set $$E_i=M_i^\dagger M_i$$, so they satisfy $$\sum_iE_i=I$$, and the probability of outcome $$i$$ is $$p_i=\langle\phi|E_i|\phi\rangle$$, but the formalism is deliberately chosen so you cannot answer the question of what the post-measurement state is.