I got a little bit confused about the definition of geodesic for $\rm SO(3)$ as

- a compact Lie group
- a Riemannian symmetric space

In the former case, it is given by the usual matrix exponential: $$ \exp_{g}tX=ge^{tX}\quad g\in{\rm SO(3)}, X\in\mathfrak{so}(3) $$ In the latter case, the geodesic is given by the transvections $\tau(\exp t(X',-X'))$: $$ {\rm Exp}_{g}~t(\frac{1}{2}X'g+\frac{1}{2}gX')=\tau(\exp\frac{t}{2}(X',-X'),g)=e^{tX'/2}ge^{tX'/2} $$ where $\tau$ is given by: $$ \begin{aligned} \tau:{(\rm SO(3)\times SO(3))}\times{\rm SO(3)}&\to{\rm SO(3)}\\ ((g,h),x)&\mapsto gxh^{-1} \end{aligned} $$ The two only coincide at the identity. Now my question is, we have two types of geodesics, all defined by the same bi-invariant metric (though in difference sense). Is there anything wrong with my calculation?

If the two types of geodesics are indeed different, which one will be shorter in length?