  Trisection maps are certain stable maps from closed $4$--manifolds to
$\mathbf{R}^2$. A simpler but reasonable class of trisection maps was
introduced by Baykur and Saeki, called a simplified $(g, k)$-trisection. We
focus on the right-left equivalence classes of simplified $(2, 0)$-trisections.
Simplified trisections are determined by their simplified trisection diagrams,
which are diagrams on a genus-$2$ surface consisting of simple closed curves of
vanishig cycles with labels. The aim of this paper is to study how the
replacement of reference paths changes simplified trisection diagrams up to
upper-triangular handle-slides. We show that for a simplified trisection $f$
satisfying a certain condition, there exists at least two simplified $(2,
0)$-trisections $f'$ and $f''$ such that $f, f'$ and $f''$ are right-left
equivalent to each other but their simplified trisection diagrams are not
related by automorphisms of a genus-$2$ surface and upper-triangular
handle-slides.
