  In recent work [PRE 100, 022121 (2019)] evidence was found that the surface
adsorption transition of interacting self-avoiding trails (ISATs) placed on the
square lattice displays a non-universal behavior at the special adsorption
point (SAP) where the collapsing polymers adsorb. In fact, different surface
exponents $\phi^{(s)}$ and $1/\delta^{(s)}$ were found at the SAP depending on
whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we
revisit these systems and study other ones, through extensive Monte Carlo
simulations, considering much longer trails than previous works. Importantly,
we demonstrate that the different exponents observed in the reference above are
due to the presence of a previously unseen surface-attached-globule (SAG) phase
in the DS system, which changes the multicritical nature of the SAP and is
absent in the HS case. By considering a modified horizontal surface (mHS) where
the trails are forbidden of having two consecutive steps along it, resembling
the DS situation, a stable SAG phase is found in the phase diagram, and both DS
and mHS systems present similar $1/\delta^{(s)}$ exponents at the SAP, being
$1/\delta^{(s)} \approx 0.44$, whilst $1/\delta^{(s)} \approx 0.34$ in the HS
case. Intriguingly, while $\phi^{(s)} \approx 1/\delta^{(s)}$ is found for the
DS and HS scenarios, as expected, in the mHS case $\phi^{(s)}$ is about $10$\%
smaller than $1/\delta^{(s)}$. These results strongly indicate that at least
two universality classes exist for the SAPs of adsorbing ISATs on the square
lattice.
