Let
g
\mathfrak {g}
be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let
μ
\mu
be a diagram automorphism of
g
\mathfrak {g}
, and let
L
(
g
,
μ
)
\mathcal {L}(\mathfrak {g},\mu )
be the loop algebra of
g
\mathfrak {g}
associated to
μ
\mu
. In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension
g
^
[
μ
]
\widehat {\mathfrak {g}}[\mu ]
of
L
(
g
,
μ
)
\mathcal {L}(\mathfrak {g},\mu )
. The construction contains the classical limit of Drinfeld’s new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212–216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283–307] as special examples. As an application, when
g
\mathfrak {g}
is of simply-laced-type, we prove that the classical limit of the
μ
\mu
-twisted quantum affinization of the quantum Kac-Moody algebra associated to
g
\mathfrak {g}
introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of
g
^
[
μ
]
\widehat {\mathfrak {g}}[\mu ]
.