  In this study, we first establish that gravity models incorporating
matter-related terms, such as $f(\mathcal{L}_{\rm m})$, $f(g_{\mu\nu}
T^{\mu\nu})$, and $f(T_{\mu\nu} T^{\mu\nu})$, into the usual matter Lagrangian
density $\mathcal{L}_{\rm m}$, are equivalent to general relativity with
nonminimal matter interactions. Through the redefinition $\mathcal{L}_{\rm m}+f
\rightarrow \mathcal{L}_{\rm m}^{\rm tot}$, these models are exactly GR, yet
the usual material field $T_{\mu\nu}$ and its accompanying partner, the
modification field $T_{\mu\nu}^{\rm mod}$, engage in nonminimal interactions.
Specifically, $\nabla^{\mu}T_{\mu\nu}=-Q_{\nu}=-\nabla^{\mu}T_{\mu\nu}^{\rm
mod}$, where $Q_{\nu}$ is the interaction kernel that governs the rate of
energy transfer. Our focus narrows on the specific model of $f(T_{\mu\nu}
T^{\mu\nu})$, known as Energy-Momentum Squared Gravity, where the usual
material field $T_{\mu\nu}$ is accompanied by an \textit{energy-momentum
squared field} (EMSF), $T_{\mu\nu}^{\rm emsf}$, along with a sui generis
nonminimal interaction between them. We demonstrate that a particular
$T_{\mu\nu}^{\rm emsf}$ can be introduced by \textit{removing}
$\frac{\partial^2 \mathcal{L}_{\rm m}}{\partial g^{\mu\nu} \partial
g^{\sigma\epsilon}}$ (the new term emerging in models that incorporate scalars
formed from $T_{\mu\nu}$), thanks to the freedom in determining the interaction
kernel, but this approach compromises the Lagrangian formulation of EMSG.
Additionally, we address the ambiguities regarding the perfect fluid stemming
from this new term. We show the proper way of calculating this term for a
perfect fluid, revealing that it is indeed non-zero, contrary to common
assumption in the literature. Finally, we re-examine cosmological models within
the realm of EMSG, offering new insights into the applicability and
interpretation of our findings in EMSG and similar theoretical frameworks.
