What is correct definition of tangential accleration?

I am asking this because i am sort of confused because there is no tangential acceleration in uniform circular motion. If this is the case, then we are considering the rate of change of magnitude of velocity, not both magnitude and direction.

The clue is in the word 'uniform': it means that angular velocity $omega$ and tangential velocity $v$ are constant in magnitude (but not direction), with the relation:

$$v=omega R$$

where $R$ is the radius of the circle.

However, there is acceleration because the velocity vector $mathbf{v}$ constantly changes direction. The so-called centripetal acceleration $mathbf{a_c}$ is given by:

$$mathbf{a_c}=frac{text{d}mathbf{v}}{text{d}t}$$

Or:

$$a_c=frac{v^2}{R}$$

This acceleration vector points perpendicularly towards the centre of the circle and keeps the object on its uniform circular trajectory.

In uniform circular motion, the tangential velocity is constant in magnitude, but its direction keeps on changing. This is because the acceleration is directed inwards to the center of the trajectory. The acceleration vector and tangential velocity vector are at right angles to each other.

There is no change of magnitude of velocity in uniform circular motion, only change in direction. Tangential acceleration is present if angular acceleration is not $0$. $$a_t=ralpha$$ where $r$ is radius of path and $alpha$ is angular acceleration. Sal explains it nicely in this video.

You can define two components of acceleration begin{equation} vec a = frac{dvec v}{dt} end{equation} parallel to velocity (tangential) begin{equation} vec a_parallel = frac{vec v , (vec a cdotvec v)}{vec v^2}, end{equation} and orthogonal to velocitybegin{equation} vec a_perp = vec a - vec a_parallel end{equation}