In their book Galois Theories, Borceux and Janelidze define a precategory internal to $\mathsf C $ to be a diagram in $\mathsf C$:
satisfying the identities
$$d_1f_0=d_0f_1,\; d_0n=1,\;d_1n=1,\;d_1f_1=d_1m,\;d_0f_0=d_0m.$$
This is the same as relaxing the pullback condition to mere commutativity in the definition of an internal category. Alternatively, one could view this as a truncated simplicial object.
How should one think of $P_2$?
The usual pullback condition means $P_2$ should be thought of as the object of composable pairs of arrows. The nlab only mentions precategories as special cases of "paracategories" but I don't really understand what that's about since their $C_2$ is defined in terms of iterated pullbacks.
$P_2$ is indeed the object of composable pairs (with chosen composition.) An "element" $x$ of $P_2$ has three arrows $f_0x,f_1x,mx$ such that the source and target of the outer arrows match-that's your first axiom-and such that the middle arrow or composition $mx$ has the appropriate domain and codomain-that's your last two axioms. This aligns decently with the nLab's definition of a precategory as a paracategory generated by binary compositions. If the usual pullback exists, $P_2$ maps into it; if this mapping is monic then $m$ does become a partially defined composition, at least in a sufficiently regular category. This map will only be monic if $P_2$ maps monically into $P_1\times P_1$, again assuming the product exists, But there's nothing in this definition to suggest this should be so. In particular, this definition permits multiple compositions for a given choice of arrows.