  We study a refinement of the symmetric multiple zeta value, called the
$t$-adic symmetric multiple zeta value, by considering its finite truncation.
More precisely, two kinds of regularizations (harmonic and shuffle) give two
kinds of the $t$-adic symmetric multiple zeta values, thus we introduce two
kinds of truncations correspondingly. Then we show that our truncations tend to
the corresponding $t$-adic symmetric multiple zeta values, and satisfy the
harmonic and shuffle relations, respectively. This gives a new proof of the
double shuffle relations for $t$-adic symmetric multiple zeta values, first
proved by Jarossay. In order to prove the shuffle relation, we develop the
theory of truncated $t$-adic symmetric multiple zeta values associated with
$2$-colored rooted trees. Finally, we discuss a refinement of Kaneko-Zagier's
conjecture and the $t$-adic symmetric multiple zeta values of Mordell-Tornheim
type.
