Projective object characterisations

Assume $P$ is projective. Then [L1] gives a lift of every map $P\to M$ across every epimorphism $E\twoheadrightarrow M$, so the last map in condition 2 is surjective. Together with the left exactness in [F1], this proves condition 2.

Condition 2 clearly implies condition 1, because surjectivity of $\mathcal{A}(P,E)\to \mathcal{A}(P,M)$ for every short exact sequence is exactly the lifting property [L1].

If $P$ is projective and $q:E\twoheadrightarrow P$ is epic, apply [L1] to $1_P:P\to P$. A lift $s:P\to E$ with $qs=1_P$ is a section, so $q$ splits.

Assume condition 3. Given an epimorphism $q:E\twoheadrightarrow M$ and a map $f:P\to M$, form the pullback of $q$ along $f$. By [L2], its projection to $P$ is epic, so condition 3 makes it split. Composing such a section with the other pullback leg gives a lift of $f$ across $q$. Thus $P$ is projective.

Steps 1.1 and 1.2 prove $1\iff 2$, and steps 1.3 and 1.4 prove $1\iff 3$. Hence all three conditions are equivalent.