A counterexample is given by $$A=\begin{bmatrix} -1 & a \\ a & -1 \\ \end{bmatrix}$$ with (say) $a=2$.
Indeed, then the sum of your first series is $$c(t):=\frac12\begin{bmatrix} c_+(t) & c_-t) \\ c_-(t) & c_+(t) \\ \end{bmatrix}$$ with $$c_\pm(t):=\cosh t\pm\cos \left(\sqrt{3} t\right)\ge1-1=0$$ for real $t$, and the sum of your second series is $$s(t):=\frac12\begin{bmatrix} s_+(t) & s_-t) \\ s_-(t) & s_+(t) \\ \end{bmatrix}$$ with $$s_\pm(t):=\sinh t\pm\frac{\sin \left(\sqrt{3} t\right)}{\sqrt{3}}\ge0$$ for real $t\ge0$, because $s_\pm(0)=0$ and $s'_\pm=c_\pm\ge0$.