I saw in the answer of this post that any finitely generated monoids are finitely presented in the sense that there is a coequalizer diagram $P_1\rightrightarrows P_0\rightarrow M$ with $P_1$ and $P_0$ free commutative and finitely generated.

My question is:

Can we make $P_1$ maps to the kernel of $P_0\rightarrow M$? (to be more, precise, can we find a presentation of the form $P_1\rightarrow P_0\rightarrow M$ like the case for abelian groups?)

And if a monoid is finitely presented for one presentation, is it the same for other presentations? Thanks