A simple question, just for clarifying: suppose we have two riemannian metrics $g$ and $\tilde{g}$ in a differentiable manifold $M$, and assume they are conformal say, with $\tilde{g} = \mu g$ for some positive valued differentiable function $\mu : M \to \mathbb{R}$. This means that
$$\tilde{g}(p)(v, w) = \mu(p) g(p)(v, w), \quad \forall p \in M, \forall v,w \in T_p M$$
right?
Yes indeed! Your definition is correct.
Just to add a little meat ... the intuition here is that the function $\mu$ tells you how much the scale in each tangent space changes. If $\mu$ is large at $p$, then a circle of radius $1$ with respect to $g$ has large radius with respect to $\widetilde{g}$, and if $\mu$ is small, then a circle of radius $1$ with respect to $g$ has correspondingly small radius with respect to $\widetilde{g}$.
Note, however, that circles are still circles. The only difference is pointwise volume; angles remain the same.