  This paper is a continuation of a previous one [Jancovici and Samaj, 2004 J.
Stat. Mech. P08006] dealing with classical Casimir phenomena in semi-infinite
wall geometries. In that paper, using microscopic Coulomb systems, the
long-ranged Casimir force due to thermal fluctuations in conducting walls was
shown to be screened by the presence of an electrolyte between the walls into
some residual short-ranged force. Here, we aim to extend the study of the
screening (cancellation) phenomena to universal Casimir terms appearing in the
large-size expansions of the grand potentials for microscopic Coulomb systems
confined in fully-finite 2D geometries, in particular the disc geometry. Two
cases are solved exactly: the high-temperature (Debye-H\"uckel) limit and the
Thirring free-fermion point. Similarities and fundamental differences between
fully-finite and semi-infinite geometries are pointed out.
