In Comm. Algebra 30(3) (2002), 1475 - 1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act $A_{S}$ over a monoid $S$ that are based on the extent to which the functor $A_{S}\otimes -$ preserves equalizers. In Semigroup Forum 65 (3) (2002), 428--449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act $A_{S}$ is called weakly equalizer-flat if the functor $A_{S}\otimes -$ 'almost' preserves equalizers of any two homomorphisms into the left act $_{S}S,$ and strongly torsion-free if this functor 'almost' preserves equalizers of any two homomorphisms from $_{S}S$ into the Rees factor act $_{S}(S/Sc),$ where $c$ is any right cancellable element of $S$. (The adverb 'almost' signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizer-flat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.