I've found two definitions for the velocity of a transverse wave on a stretched string: $$ \begin{align} v & = \frac{\omega}{k} \tag{1} \\[10px] v & = \sqrt{\frac{T}{u}} \tag{2} \end{align} $$
Question: Are these two definitions mutually consistent, or is there a fundamental difference?
I'm asking because $\operatorname{Eq.}{\left(1\right)}$ suggests dependence on frequency while $\operatorname{Eq.}{\left(2\right)}$ doesn't.
The definition of something is what it means. The definition of the speed of a (sinusoidal) wave is the distance a wavefront moves per unit time.
Your first formula, $v=\frac{\omega}{k}$, is closely related to the definition of $v$ (though not strictly the definition). The presence of $\omega$ doesn't necessarily indicate a dependence on frequency, because $k$ may well depend on frequency (even though the definition of $k$ doesn't involve frequency!). In fact, for many types of wave the speed is independent of the frequency, so $k$ is proportional to $\omega$.
Your second formula, $v=\sqrt{\frac{T}{\mu}}$, relates the speed of the wave on a stretched string to the tension in the string and the string's mass per unit length. It is certainly not a definition of the speed of the wave. As you say, this formula implies that the speed of a transverse wave on a stretched string doesn't depend on the frequency, because we know that neither $T$ nor $\mu$ has any dependence on frequency. [The formula only holds as long as certain conditions apply, for example the wave amplitude is much less than the wavelength.]
